Theorem dfac12lem2 8849

Description: Lemma for dfac12 8854. (Contributed by Mario Carneiro, 29-May-2015.)

Hypotheses

Ref	Expression
dfac12.1	⊢ (𝜑 → 𝐴 ∈ On)
dfac12.3	⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On)
dfac12.4	⊢ 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))))))
dfac12.5	⊢ (𝜑 → 𝐶 ∈ On)
dfac12.h	⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶))
dfac12.6	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
dfac12.8	⊢ (𝜑 → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)

Assertion

Ref	Expression
dfac12lem2	⊢ (𝜑 → (𝐺‘𝐶):(𝑅1‘𝐶)–1-1→On)

Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑦,𝑧,𝐶   𝑥,𝐺,𝑦,𝑧   𝜑,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑦,𝐻,𝑧

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐻(𝑥)

Proof of Theorem dfac12lem2

Step	Hyp	Ref	Expression
1		dfac12.4	. . . . . . . . . . . . . 14 ⊢ 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))))))
2	1	tfr1 7380	. . . . . . . . . . . . 13 ⊢ 𝐺 Fn On
3		fnfun 5902	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn On → Fun 𝐺)
4	2, 3	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 𝐺
5		dfac12.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ On)
6		funimaexg 5889	. . . . . . . . . . . 12 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 “ 𝐶) ∈ V)
7	4, 5, 6	sylancr 694	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 “ 𝐶) ∈ V)
8		uniexg 6853	. . . . . . . . . . 11 ⊢ ((𝐺 “ 𝐶) ∈ V → ∪ (𝐺 “ 𝐶) ∈ V)
9		rnexg 6990	. . . . . . . . . . 11 ⊢ (∪ (𝐺 “ 𝐶) ∈ V → ran ∪ (𝐺 “ 𝐶) ∈ V)
10	7, 8, 9	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ran ∪ (𝐺 “ 𝐶) ∈ V)
11		dfac12.8	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
12		f1f 6014	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On → (𝐺‘𝑧):(𝑅1‘𝑧)⟶On)
13		fssxp 5973	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑧):(𝑅1‘𝑧)⟶On → (𝐺‘𝑧) ⊆ ((𝑅1‘𝑧) × On))
14		ssv 3588	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅1‘𝑧) ⊆ V
15		xpss1 5151	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅1‘𝑧) ⊆ V → ((𝑅1‘𝑧) × On) ⊆ (V × On))
16	14, 15	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅1‘𝑧) × On) ⊆ (V × On)
17		sstr 3576	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺‘𝑧) ⊆ ((𝑅1‘𝑧) × On) ∧ ((𝑅1‘𝑧) × On) ⊆ (V × On)) → (𝐺‘𝑧) ⊆ (V × On))
18	16, 17	mpan2 703	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑧) ⊆ ((𝑅1‘𝑧) × On) → (𝐺‘𝑧) ⊆ (V × On))
19		fvex 6113	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺‘𝑧) ∈ V
20	19	elpw 4114	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑧) ∈ 𝒫 (V × On) ↔ (𝐺‘𝑧) ⊆ (V × On))
21	18, 20	sylibr 223	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑧) ⊆ ((𝑅1‘𝑧) × On) → (𝐺‘𝑧) ∈ 𝒫 (V × On))
22	12, 13, 21	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On → (𝐺‘𝑧) ∈ 𝒫 (V × On))
23	22	ralimi 2936	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧) ∈ 𝒫 (V × On))
24	11, 23	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧) ∈ 𝒫 (V × On))
25		onss 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
26	5, 25	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ⊆ On)
27		fndm 5904	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 Fn On → dom 𝐺 = On)
28	2, 27	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ dom 𝐺 = On
29	26, 28	syl6sseqr 3615	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ⊆ dom 𝐺)
30		funimass4 6157	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐺 ∧ 𝐶 ⊆ dom 𝐺) → ((𝐺 “ 𝐶) ⊆ 𝒫 (V × On) ↔ ∀𝑧 ∈ 𝐶 (𝐺‘𝑧) ∈ 𝒫 (V × On)))
31	4, 29, 30	sylancr 694	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐺 “ 𝐶) ⊆ 𝒫 (V × On) ↔ ∀𝑧 ∈ 𝐶 (𝐺‘𝑧) ∈ 𝒫 (V × On)))
32	24, 31	mpbird 246	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 “ 𝐶) ⊆ 𝒫 (V × On))
33		sspwuni 4547	. . . . . . . . . . . . 13 ⊢ ((𝐺 “ 𝐶) ⊆ 𝒫 (V × On) ↔ ∪ (𝐺 “ 𝐶) ⊆ (V × On))
34	32, 33	sylib 207	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ (𝐺 “ 𝐶) ⊆ (V × On))
35		rnss 5275	. . . . . . . . . . . 12 ⊢ (∪ (𝐺 “ 𝐶) ⊆ (V × On) → ran ∪ (𝐺 “ 𝐶) ⊆ ran (V × On))
36	34, 35	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ (𝐺 “ 𝐶) ⊆ ran (V × On))
37		rnxpss 5485	. . . . . . . . . . 11 ⊢ ran (V × On) ⊆ On
38	36, 37	syl6ss 3580	. . . . . . . . . 10 ⊢ (𝜑 → ran ∪ (𝐺 “ 𝐶) ⊆ On)
39		ssonuni 6878	. . . . . . . . . 10 ⊢ (ran ∪ (𝐺 “ 𝐶) ∈ V → (ran ∪ (𝐺 “ 𝐶) ⊆ On → ∪ ran ∪ (𝐺 “ 𝐶) ∈ On))
40	10, 38, 39	sylc 63	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran ∪ (𝐺 “ 𝐶) ∈ On)
41		suceloni 6905	. . . . . . . . 9 ⊢ (∪ ran ∪ (𝐺 “ 𝐶) ∈ On → suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On)
42	40, 41	syl 17	. . . . . . . 8 ⊢ (𝜑 → suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On)
43	42	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On)
44		rankon 8541	. . . . . . 7 ⊢ (rank‘𝑦) ∈ On
45		omcl 7503	. . . . . . 7 ⊢ ((suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On ∧ (rank‘𝑦) ∈ On) → (suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) ∈ On)
46	43, 44, 45	sylancl 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) ∈ On)
47		rankr1ai 8544	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑅1‘𝐶) → (rank‘𝑦) ∈ 𝐶)
48	47	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ 𝐶)
49		simpr 476	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝐶 = ∪ 𝐶)
50	48, 49	eleqtrd 2690	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ ∪ 𝐶)
51		eloni 5650	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → Ord 𝐶)
52	5, 51	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → Ord 𝐶)
53	52	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → Ord 𝐶)
54		ordsucuniel 6916	. . . . . . . . . . 11 ⊢ (Ord 𝐶 → ((rank‘𝑦) ∈ ∪ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
55	53, 54	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((rank‘𝑦) ∈ ∪ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
56	50, 55	mpbid 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → suc (rank‘𝑦) ∈ 𝐶)
57	11	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
58		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑧 = suc (rank‘𝑦) → (𝐺‘𝑧) = (𝐺‘suc (rank‘𝑦)))
59		f1eq1 6009	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑧) = (𝐺‘suc (rank‘𝑦)) → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘suc (rank‘𝑦)):(𝑅1‘𝑧)–1-1→On))
60	58, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 = suc (rank‘𝑦) → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘suc (rank‘𝑦)):(𝑅1‘𝑧)–1-1→On))
61		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑧 = suc (rank‘𝑦) → (𝑅1‘𝑧) = (𝑅1‘suc (rank‘𝑦)))
62		f1eq2 6010	. . . . . . . . . . . 12 ⊢ ((𝑅1‘𝑧) = (𝑅1‘suc (rank‘𝑦)) → ((𝐺‘suc (rank‘𝑦)):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On))
63	61, 62	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 = suc (rank‘𝑦) → ((𝐺‘suc (rank‘𝑦)):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On))
64	60, 63	bitrd 267	. . . . . . . . . 10 ⊢ (𝑧 = suc (rank‘𝑦) → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On))
65	64	rspcv 3278	. . . . . . . . 9 ⊢ (suc (rank‘𝑦) ∈ 𝐶 → (∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On))
66	56, 57, 65	sylc 63	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On)
67		f1f 6014	. . . . . . . 8 ⊢ ((𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))⟶On)
68	66, 67	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))⟶On)
69		r1elwf 8542	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑅1‘𝐶) → 𝑦 ∈ ∪ (𝑅1 “ On))
70	69	ad2antlr 759	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝑦 ∈ ∪ (𝑅1 “ On))
71		rankidb 8546	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
72	70, 71	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
73	68, 72	ffvelrnd 6268	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ On)
74		oacl 7502	. . . . . 6 ⊢ (((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) ∈ On ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ On) → ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)) ∈ On)
75	46, 73, 74	syl2anc 691	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)) ∈ On)
76		dfac12.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On)
77		f1f 6014	. . . . . . . 8 ⊢ (𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On → 𝐹:𝒫 (har‘(𝑅1‘𝐴))⟶On)
78	76, 77	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))⟶On)
79	78	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐹:𝒫 (har‘(𝑅1‘𝐴))⟶On)
80		imassrn 5396	. . . . . . . 8 ⊢ (𝐻 “ 𝑦) ⊆ ran 𝐻
81		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝐺‘∪ 𝐶) ∈ V
82	81	rnex 6992	. . . . . . . . . . . . . 14 ⊢ ran (𝐺‘∪ 𝐶) ∈ V
83	5	ad2antrr 758	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 ∈ On)
84		onuni 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ On → ∪ 𝐶 ∈ On)
85		sucidg 5720	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝐶 ∈ On → ∪ 𝐶 ∈ suc ∪ 𝐶)
86	83, 84, 85	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ suc ∪ 𝐶)
87	52	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → Ord 𝐶)
88		orduniorsuc 6922	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Ord 𝐶 → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
89	87, 88	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
90	89	orcanai 950	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶)
91	86, 90	eleqtrrd 2691	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐶)
92	11	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On)
93		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ∪ 𝐶 → (𝐺‘𝑧) = (𝐺‘∪ 𝐶))
94		f1eq1 6009	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺‘𝑧) = (𝐺‘∪ 𝐶) → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘∪ 𝐶):(𝑅1‘𝑧)–1-1→On))
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ∪ 𝐶 → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘∪ 𝐶):(𝑅1‘𝑧)–1-1→On))
96		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ∪ 𝐶 → (𝑅1‘𝑧) = (𝑅1‘∪ 𝐶))
97		f1eq2 6010	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅1‘𝑧) = (𝑅1‘∪ 𝐶) → ((𝐺‘∪ 𝐶):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On))
98	96, 97	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ∪ 𝐶 → ((𝐺‘∪ 𝐶):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On))
99	95, 98	bitrd 267	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ∪ 𝐶 → ((𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On ↔ (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On))
100	99	rspcv 3278	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝐶 ∈ 𝐶 → (∀𝑧 ∈ 𝐶 (𝐺‘𝑧):(𝑅1‘𝑧)–1-1→On → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On))
101	91, 92, 100	sylc 63	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On)
102		f1f 6014	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)⟶On)
103		frn 5966	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)⟶On → ran (𝐺‘∪ 𝐶) ⊆ On)
104	101, 102, 103	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran (𝐺‘∪ 𝐶) ⊆ On)
105		epweon 6875	. . . . . . . . . . . . . . 15 ⊢ E We On
106		wess 5025	. . . . . . . . . . . . . . 15 ⊢ (ran (𝐺‘∪ 𝐶) ⊆ On → ( E We On → E We ran (𝐺‘∪ 𝐶)))
107	104, 105, 106	mpisyl 21	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → E We ran (𝐺‘∪ 𝐶))
108		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ OrdIso( E , ran (𝐺‘∪ 𝐶)) = OrdIso( E , ran (𝐺‘∪ 𝐶))
109	108	oiiso 8325	. . . . . . . . . . . . . 14 ⊢ ((ran (𝐺‘∪ 𝐶) ∈ V ∧ E We ran (𝐺‘∪ 𝐶)) → OrdIso( E , ran (𝐺‘∪ 𝐶)) Isom E , E (dom OrdIso( E , ran (𝐺‘∪ 𝐶)), ran (𝐺‘∪ 𝐶)))
110	82, 107, 109	sylancr 694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → OrdIso( E , ran (𝐺‘∪ 𝐶)) Isom E , E (dom OrdIso( E , ran (𝐺‘∪ 𝐶)), ran (𝐺‘∪ 𝐶)))
111		isof1o 6473	. . . . . . . . . . . . 13 ⊢ (OrdIso( E , ran (𝐺‘∪ 𝐶)) Isom E , E (dom OrdIso( E , ran (𝐺‘∪ 𝐶)), ran (𝐺‘∪ 𝐶)) → OrdIso( E , ran (𝐺‘∪ 𝐶)):dom OrdIso( E , ran (𝐺‘∪ 𝐶))–1-1-onto→ran (𝐺‘∪ 𝐶))
112		f1ocnv 6062	. . . . . . . . . . . . 13 ⊢ (OrdIso( E , ran (𝐺‘∪ 𝐶)):dom OrdIso( E , ran (𝐺‘∪ 𝐶))–1-1-onto→ran (𝐺‘∪ 𝐶) → ◡OrdIso( E , ran (𝐺‘∪ 𝐶)):ran (𝐺‘∪ 𝐶)–1-1-onto→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
113		f1of1 6049	. . . . . . . . . . . . 13 ⊢ (◡OrdIso( E , ran (𝐺‘∪ 𝐶)):ran (𝐺‘∪ 𝐶)–1-1-onto→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) → ◡OrdIso( E , ran (𝐺‘∪ 𝐶)):ran (𝐺‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
114	110, 111, 112, 113	4syl 19	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ◡OrdIso( E , ran (𝐺‘∪ 𝐶)):ran (𝐺‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
115		f1f1orn 6061	. . . . . . . . . . . . 13 ⊢ ((𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→On → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1-onto→ran (𝐺‘∪ 𝐶))
116		f1of1 6049	. . . . . . . . . . . . 13 ⊢ ((𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1-onto→ran (𝐺‘∪ 𝐶) → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→ran (𝐺‘∪ 𝐶))
117	101, 115, 116	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→ran (𝐺‘∪ 𝐶))
118		f1co 6023	. . . . . . . . . . . 12 ⊢ ((◡OrdIso( E , ran (𝐺‘∪ 𝐶)):ran (𝐺‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∧ (𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1→ran (𝐺‘∪ 𝐶)) → (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)):(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
119	114, 117, 118	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)):(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
120		dfac12.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶))
121		f1eq1 6009	. . . . . . . . . . . 12 ⊢ (𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)) → (𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ↔ (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)):(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶))))
122	120, 121	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ↔ (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)):(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
123	119, 122	sylibr 223	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
124		f1f 6014	. . . . . . . . . 10 ⊢ (𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) → 𝐻:(𝑅1‘∪ 𝐶)⟶dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
125		frn 5966	. . . . . . . . . 10 ⊢ (𝐻:(𝑅1‘∪ 𝐶)⟶dom OrdIso( E , ran (𝐺‘∪ 𝐶)) → ran 𝐻 ⊆ dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
126	123, 124, 125	3syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran 𝐻 ⊆ dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
127		harcl 8349	. . . . . . . . . . 11 ⊢ (har‘(𝑅1‘𝐴)) ∈ On
128	127	onordi 5749	. . . . . . . . . 10 ⊢ Ord (har‘(𝑅1‘𝐴))
129	108	oion 8324	. . . . . . . . . . . 12 ⊢ (ran (𝐺‘∪ 𝐶) ∈ V → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ On)
130	82, 129	mp1i 13	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ On)
131	108	oien 8326	. . . . . . . . . . . . 13 ⊢ ((ran (𝐺‘∪ 𝐶) ∈ V ∧ E We ran (𝐺‘∪ 𝐶)) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≈ ran (𝐺‘∪ 𝐶))
132	82, 107, 131	sylancr 694	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≈ ran (𝐺‘∪ 𝐶))
133		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝑅1‘∪ 𝐶) ∈ V
134	133	f1oen 7862	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘∪ 𝐶):(𝑅1‘∪ 𝐶)–1-1-onto→ran (𝐺‘∪ 𝐶) → (𝑅1‘∪ 𝐶) ≈ ran (𝐺‘∪ 𝐶))
135		ensym 7891	. . . . . . . . . . . . . 14 ⊢ ((𝑅1‘∪ 𝐶) ≈ ran (𝐺‘∪ 𝐶) → ran (𝐺‘∪ 𝐶) ≈ (𝑅1‘∪ 𝐶))
136	101, 115, 134, 135	4syl 19	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran (𝐺‘∪ 𝐶) ≈ (𝑅1‘∪ 𝐶))
137		fvex 6113	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘𝐴) ∈ V
138		dfac12.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ On)
139	138	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐴 ∈ On)
140		dfac12.6	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
141	140	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 ⊆ 𝐴)
142	141, 91	sseldd 3569	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐴)
143		r1ord2 8527	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → (∪ 𝐶 ∈ 𝐴 → (𝑅1‘∪ 𝐶) ⊆ (𝑅1‘𝐴)))
144	139, 142, 143	sylc 63	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘∪ 𝐶) ⊆ (𝑅1‘𝐴))
145		ssdomg 7887	. . . . . . . . . . . . . 14 ⊢ ((𝑅1‘𝐴) ∈ V → ((𝑅1‘∪ 𝐶) ⊆ (𝑅1‘𝐴) → (𝑅1‘∪ 𝐶) ≼ (𝑅1‘𝐴)))
146	137, 144, 145	mpsyl 66	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘∪ 𝐶) ≼ (𝑅1‘𝐴))
147		endomtr 7900	. . . . . . . . . . . . 13 ⊢ ((ran (𝐺‘∪ 𝐶) ≈ (𝑅1‘∪ 𝐶) ∧ (𝑅1‘∪ 𝐶) ≼ (𝑅1‘𝐴)) → ran (𝐺‘∪ 𝐶) ≼ (𝑅1‘𝐴))
148	136, 146, 147	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran (𝐺‘∪ 𝐶) ≼ (𝑅1‘𝐴))
149		endomtr 7900	. . . . . . . . . . . 12 ⊢ ((dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≈ ran (𝐺‘∪ 𝐶) ∧ ran (𝐺‘∪ 𝐶) ≼ (𝑅1‘𝐴)) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≼ (𝑅1‘𝐴))
150	132, 148, 149	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≼ (𝑅1‘𝐴))
151		elharval 8351	. . . . . . . . . . 11 ⊢ (dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ (har‘(𝑅1‘𝐴)) ↔ (dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ On ∧ dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ≼ (𝑅1‘𝐴)))
152	130, 150, 151	sylanbrc 695	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ (har‘(𝑅1‘𝐴)))
153		ordelss 5656	. . . . . . . . . 10 ⊢ ((Ord (har‘(𝑅1‘𝐴)) ∧ dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∈ (har‘(𝑅1‘𝐴))) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ⊆ (har‘(𝑅1‘𝐴)))
154	128, 152, 153	sylancr 694	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ⊆ (har‘(𝑅1‘𝐴)))
155	126, 154	sstrd 3578	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran 𝐻 ⊆ (har‘(𝑅1‘𝐴)))
156	80, 155	syl5ss 3579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑦) ⊆ (har‘(𝑅1‘𝐴)))
157		fvex 6113	. . . . . . . 8 ⊢ (har‘(𝑅1‘𝐴)) ∈ V
158	157	elpw2 4755	. . . . . . 7 ⊢ ((𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴)) ↔ (𝐻 “ 𝑦) ⊆ (har‘(𝑅1‘𝐴)))
159	156, 158	sylibr 223	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴)))
160	79, 159	ffvelrnd 6268	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐹‘(𝐻 “ 𝑦)) ∈ On)
161	75, 160	ifclda 4070	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) ∈ On)
162	161	ex 449	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝑅1‘𝐶) → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) ∈ On))
163		iftrue 4042	. . . . . . . 8 ⊢ (𝐶 = ∪ 𝐶 → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)))
164		iftrue 4042	. . . . . . . 8 ⊢ (𝐶 = ∪ 𝐶 → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)))
165	163, 164	eqeq12d 2625	. . . . . . 7 ⊢ (𝐶 = ∪ 𝐶 → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧))))
166	165	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧))))
167	42	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On)
168		nsuceq0 5722	. . . . . . . 8 ⊢ suc ∪ ran ∪ (𝐺 “ 𝐶) ≠ ∅
169	168	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → suc ∪ ran ∪ (𝐺 “ 𝐶) ≠ ∅)
170	44	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ On)
171		onsucuni 6920	. . . . . . . . . . 11 ⊢ (ran ∪ (𝐺 “ 𝐶) ⊆ On → ran ∪ (𝐺 “ 𝐶) ⊆ suc ∪ ran ∪ (𝐺 “ 𝐶))
172	38, 171	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ran ∪ (𝐺 “ 𝐶) ⊆ suc ∪ ran ∪ (𝐺 “ 𝐶))
173	172	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ran ∪ (𝐺 “ 𝐶) ⊆ suc ∪ ran ∪ (𝐺 “ 𝐶))
174	2	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝐺 Fn On)
175	26	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝐶 ⊆ On)
176		fnfvima 6400	. . . . . . . . . . . 12 ⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On ∧ suc (rank‘𝑦) ∈ 𝐶) → (𝐺‘suc (rank‘𝑦)) ∈ (𝐺 “ 𝐶))
177	174, 175, 56, 176	syl3anc 1318	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (𝐺‘suc (rank‘𝑦)) ∈ (𝐺 “ 𝐶))
178		elssuni 4403	. . . . . . . . . . 11 ⊢ ((𝐺‘suc (rank‘𝑦)) ∈ (𝐺 “ 𝐶) → (𝐺‘suc (rank‘𝑦)) ⊆ ∪ (𝐺 “ 𝐶))
179		rnss 5275	. . . . . . . . . . 11 ⊢ ((𝐺‘suc (rank‘𝑦)) ⊆ ∪ (𝐺 “ 𝐶) → ran (𝐺‘suc (rank‘𝑦)) ⊆ ran ∪ (𝐺 “ 𝐶))
180	177, 178, 179	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ran (𝐺‘suc (rank‘𝑦)) ⊆ ran ∪ (𝐺 “ 𝐶))
181		f1fn 6015	. . . . . . . . . . . 12 ⊢ ((𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On → (𝐺‘suc (rank‘𝑦)) Fn (𝑅1‘suc (rank‘𝑦)))
182	66, 181	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (𝐺‘suc (rank‘𝑦)) Fn (𝑅1‘suc (rank‘𝑦)))
183		fnfvelrn 6264	. . . . . . . . . . 11 ⊢ (((𝐺‘suc (rank‘𝑦)) Fn (𝑅1‘suc (rank‘𝑦)) ∧ 𝑦 ∈ (𝑅1‘suc (rank‘𝑦))) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ ran (𝐺‘suc (rank‘𝑦)))
184	182, 72, 183	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ ran (𝐺‘suc (rank‘𝑦)))
185	180, 184	sseldd 3569	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ ran ∪ (𝐺 “ 𝐶))
186	173, 185	sseldd 3569	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))
187	186	adantlrr 753	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))
188		rankon 8541	. . . . . . . 8 ⊢ (rank‘𝑧) ∈ On
189	188	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑧) ∈ On)
190		eleq1 2676	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1‘𝐶) ↔ 𝑧 ∈ (𝑅1‘𝐶)))
191	190	anbi2d 736	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ↔ (𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶))))
192	191	anbi1d 737	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) ↔ ((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶)))
193		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
194		suceq 5707	. . . . . . . . . . . . . 14 ⊢ ((rank‘𝑦) = (rank‘𝑧) → suc (rank‘𝑦) = suc (rank‘𝑧))
195	193, 194	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → suc (rank‘𝑦) = suc (rank‘𝑧))
196	195	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝐺‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑧)))
197		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
198	196, 197	fveq12d 6109	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧))
199	198	eleq1d 2672	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶) ↔ ((𝐺‘suc (rank‘𝑧))‘𝑧) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶)))
200	192, 199	imbi12d 333	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶)) ↔ (((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑧))‘𝑧) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))))
201	200, 186	chvarv 2251	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑧))‘𝑧) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))
202	201	adantlrl 752	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → ((𝐺‘suc (rank‘𝑧))‘𝑧) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))
203		omopth2 7551	. . . . . . 7 ⊢ (((suc ∪ ran ∪ (𝐺 “ 𝐶) ∈ On ∧ suc ∪ ran ∪ (𝐺 “ 𝐶) ≠ ∅) ∧ ((rank‘𝑦) ∈ On ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶)) ∧ ((rank‘𝑧) ∈ On ∧ ((𝐺‘suc (rank‘𝑧))‘𝑧) ∈ suc ∪ ran ∪ (𝐺 “ 𝐶))) → (((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)) ↔ ((rank‘𝑦) = (rank‘𝑧) ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧))))
204	167, 169, 170, 187, 189, 202, 203	syl222anc 1334	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)) ↔ ((rank‘𝑦) = (rank‘𝑧) ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧))))
205	194	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → suc (rank‘𝑦) = suc (rank‘𝑧))
206	205	fveq2d 6107	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (𝐺‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑧)))
207	206	fveq1d 6105	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → ((𝐺‘suc (rank‘𝑦))‘𝑧) = ((𝐺‘suc (rank‘𝑧))‘𝑧))
208	207	eqeq2d 2620	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑧) ↔ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧)))
209	66	adantlrr 753	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On)
210	209	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On)
211	72	adantlrr 753	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
212	211	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
213		r1elwf 8542	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝑅1‘𝐶) → 𝑧 ∈ ∪ (𝑅1 “ On))
214		rankidb 8546	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → 𝑧 ∈ (𝑅1‘suc (rank‘𝑧)))
215	213, 214	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑅1‘𝐶) → 𝑧 ∈ (𝑅1‘suc (rank‘𝑧)))
216	215	ad2antll 761	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) → 𝑧 ∈ (𝑅1‘suc (rank‘𝑧)))
217	216	ad2antrr 758	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → 𝑧 ∈ (𝑅1‘suc (rank‘𝑧)))
218	205	fveq2d 6107	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc (rank‘𝑧)))
219	217, 218	eleqtrrd 2691	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → 𝑧 ∈ (𝑅1‘suc (rank‘𝑦)))
220		f1fveq 6420	. . . . . . . . . . 11 ⊢ (((𝐺‘suc (rank‘𝑦)):(𝑅1‘suc (rank‘𝑦))–1-1→On ∧ (𝑦 ∈ (𝑅1‘suc (rank‘𝑦)) ∧ 𝑧 ∈ (𝑅1‘suc (rank‘𝑦)))) → (((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑧) ↔ 𝑦 = 𝑧))
221	210, 212, 219, 220	syl12anc 1316	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑧) ↔ 𝑦 = 𝑧))
222	208, 221	bitr3d 269	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧) ↔ 𝑦 = 𝑧))
223	222	biimpd 218	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) ∧ (rank‘𝑦) = (rank‘𝑧)) → (((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧) → 𝑦 = 𝑧))
224	223	expimpd 627	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (((rank‘𝑦) = (rank‘𝑧) ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧)) → 𝑦 = 𝑧))
225	193, 198	jca 553	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((rank‘𝑦) = (rank‘𝑧) ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧)))
226	224, 225	impbid1 214	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (((rank‘𝑦) = (rank‘𝑧) ∧ ((𝐺‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑧))‘𝑧)) ↔ 𝑦 = 𝑧))
227	166, 204, 226	3bitrd 293	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ 𝐶 = ∪ 𝐶) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ 𝑦 = 𝑧))
228		iffalse 4045	. . . . . . . 8 ⊢ (¬ 𝐶 = ∪ 𝐶 → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = (𝐹‘(𝐻 “ 𝑦)))
229		iffalse 4045	. . . . . . . 8 ⊢ (¬ 𝐶 = ∪ 𝐶 → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) = (𝐹‘(𝐻 “ 𝑧)))
230	228, 229	eqeq12d 2625	. . . . . . 7 ⊢ (¬ 𝐶 = ∪ 𝐶 → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ (𝐹‘(𝐻 “ 𝑦)) = (𝐹‘(𝐻 “ 𝑧))))
231	230	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ (𝐹‘(𝐻 “ 𝑦)) = (𝐹‘(𝐻 “ 𝑧))))
232	76	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On)
233	159	adantlrr 753	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴)))
234	191	anbi1d 737	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) ↔ ((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶)))
235		imaeq2 5381	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝐻 “ 𝑦) = (𝐻 “ 𝑧))
236	235	eleq1d 2672	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴)) ↔ (𝐻 “ 𝑧) ∈ 𝒫 (har‘(𝑅1‘𝐴))))
237	234, 236	imbi12d 333	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴))) ↔ (((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑧) ∈ 𝒫 (har‘(𝑅1‘𝐴)))))
238	237, 159	chvarv 2251	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑧) ∈ 𝒫 (har‘(𝑅1‘𝐴)))
239	238	adantlrl 752	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐻 “ 𝑧) ∈ 𝒫 (har‘(𝑅1‘𝐴)))
240		f1fveq 6420	. . . . . . 7 ⊢ ((𝐹:𝒫 (har‘(𝑅1‘𝐴))–1-1→On ∧ ((𝐻 “ 𝑦) ∈ 𝒫 (har‘(𝑅1‘𝐴)) ∧ (𝐻 “ 𝑧) ∈ 𝒫 (har‘(𝑅1‘𝐴)))) → ((𝐹‘(𝐻 “ 𝑦)) = (𝐹‘(𝐻 “ 𝑧)) ↔ (𝐻 “ 𝑦) = (𝐻 “ 𝑧)))
241	232, 233, 239, 240	syl12anc 1316	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐹‘(𝐻 “ 𝑦)) = (𝐹‘(𝐻 “ 𝑧)) ↔ (𝐻 “ 𝑦) = (𝐻 “ 𝑧)))
242	123	adantlrr 753	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)))
243		simplrl 796	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑦 ∈ (𝑅1‘𝐶))
244	90	fveq2d 6107	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘𝐶) = (𝑅1‘suc ∪ 𝐶))
245		r1suc 8516	. . . . . . . . . . . 12 ⊢ (∪ 𝐶 ∈ On → (𝑅1‘suc ∪ 𝐶) = 𝒫 (𝑅1‘∪ 𝐶))
246	83, 84, 245	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘suc ∪ 𝐶) = 𝒫 (𝑅1‘∪ 𝐶))
247	244, 246	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘𝐶) = 𝒫 (𝑅1‘∪ 𝐶))
248	247	adantlrr 753	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝑅1‘𝐶) = 𝒫 (𝑅1‘∪ 𝐶))
249	243, 248	eleqtrd 2690	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑦 ∈ 𝒫 (𝑅1‘∪ 𝐶))
250	249	elpwid 4118	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑦 ⊆ (𝑅1‘∪ 𝐶))
251		simplrr 797	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑧 ∈ (𝑅1‘𝐶))
252	251, 248	eleqtrd 2690	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑧 ∈ 𝒫 (𝑅1‘∪ 𝐶))
253	252	elpwid 4118	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝑧 ⊆ (𝑅1‘∪ 𝐶))
254		f1imaeq 6423	. . . . . . 7 ⊢ ((𝐻:(𝑅1‘∪ 𝐶)–1-1→dom OrdIso( E , ran (𝐺‘∪ 𝐶)) ∧ (𝑦 ⊆ (𝑅1‘∪ 𝐶) ∧ 𝑧 ⊆ (𝑅1‘∪ 𝐶))) → ((𝐻 “ 𝑦) = (𝐻 “ 𝑧) ↔ 𝑦 = 𝑧))
255	242, 250, 253, 254	syl12anc 1316	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐻 “ 𝑦) = (𝐻 “ 𝑧) ↔ 𝑦 = 𝑧))
256	231, 241, 255	3bitrd 293	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) ∧ ¬ 𝐶 = ∪ 𝐶) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ 𝑦 = 𝑧))
257	227, 256	pm2.61dan 828	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶))) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ 𝑦 = 𝑧))
258	257	ex 449	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝑅1‘𝐶) ∧ 𝑧 ∈ (𝑅1‘𝐶)) → (if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑧)) +𝑜 ((𝐺‘suc (rank‘𝑧))‘𝑧)), (𝐹‘(𝐻 “ 𝑧))) ↔ 𝑦 = 𝑧)))
259	162, 258	dom2lem 7881	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))):(𝑅1‘𝐶)–1-1→On)
260	138, 76, 1, 5, 120	dfac12lem1 8848	. . 3 ⊢ (𝜑 → (𝐺‘𝐶) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))))
261		f1eq1 6009	. . 3 ⊢ ((𝐺‘𝐶) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))) → ((𝐺‘𝐶):(𝑅1‘𝐶)–1-1→On ↔ (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))):(𝑅1‘𝐶)–1-1→On))
262	260, 261	syl 17	. 2 ⊢ (𝜑 → ((𝐺‘𝐶):(𝑅1‘𝐶)–1-1→On ↔ (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))):(𝑅1‘𝐶)–1-1→On))
263	259, 262	mpbird 246	1 ⊢ (𝜑 → (𝐺‘𝐶):(𝑅1‘𝐶)–1-1→On)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   E cep 4947   We wwe 4996   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  Ord word 5639  Oncon0 5640  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549  recscrecs 7354   +_𝑜 coa 7444   ·_𝑜 comu 7445   ≈ cen 7838   ≼ cdom 7839  OrdIsocoi 8297  harchar 8344  𝑅₁cr1 8508  rankcrnk 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452  df-er 7629  df-en 7842  df-dom 7843  df-oi 8298  df-har 8346  df-r1 8510  df-rank 8511

This theorem is referenced by:  dfac12lem3  8850