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Theorem cnfcom 8480

Description: Any ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)

**Hypotheses**
Ref	Expression
cnfcom.s	⊢ 𝑆 = dom (ω CNF 𝐴)
cnfcom.a	⊢ (𝜑 → 𝐴 ∈ On)
cnfcom.b	⊢ (𝜑 → 𝐵 ∈ (ω ↑_𝑜 𝐴))
cnfcom.f	⊢ 𝐹 = (^◡(ω CNF 𝐴)‘𝐵)
cnfcom.g	⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
cnfcom.h	⊢ 𝐻 = seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +_𝑜 𝑧)), ∅)
cnfcom.t	⊢ 𝑇 = seq_𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
cnfcom.m	⊢ 𝑀 = ((ω ↑_𝑜 (𝐺‘𝑘)) ·_𝑜 (𝐹‘(𝐺‘𝑘)))
cnfcom.k	⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +_𝑜 𝑥)) ∪ ^◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +_𝑜 𝑥)))
cnfcom.1	⊢ (𝜑 → 𝐼 ∈ dom 𝐺)

**Assertion**
Ref	Expression
cnfcom	⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝐼)) ·_𝑜 (𝐹‘(𝐺‘𝐼))))

Distinct variable groups: 𝑥,𝑘,𝑧,𝐴 𝑘,𝐼,𝑥,𝑧 𝑥,𝑀 𝑓,𝑘,𝑥,𝑧,𝐹 𝑧,𝑇 𝑓,𝐺,𝑘,𝑥,𝑧 𝑓,𝐻,𝑥 𝑆,𝑘,𝑧 Allowed substitution hints: 𝜑(𝑥,𝑧,𝑓,𝑘) 𝐴(𝑓) 𝐵(𝑥,𝑧,𝑓,𝑘) 𝑆(𝑥,𝑓) 𝑇(𝑥,𝑓,𝑘) 𝐻(𝑧,𝑘) 𝐼(𝑓) 𝐾(𝑥,𝑧,𝑓,𝑘) 𝑀(𝑧,𝑓,𝑘)

Proof of Theorem cnfcom Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnfcom.1	. 2 ⊢ (𝜑 → 𝐼 ∈ dom 𝐺)
2		cnfcom.s	. . . . . 6 ⊢ 𝑆 = dom (ω CNF 𝐴)
3		omelon 8426	. . . . . . 7 ⊢ ω ∈ On
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → ω ∈ On)
5		cnfcom.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On)
6		cnfcom.g	. . . . . 6 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
7		cnfcom.f	. . . . . . 7 ⊢ 𝐹 = (^◡(ω CNF 𝐴)‘𝐵)
8	2, 4, 5	cantnff1o 8476	. . . . . . . . 9 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑_𝑜 𝐴))
9		f1ocnv 6062	. . . . . . . . 9 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑_𝑜 𝐴) → ^◡(ω CNF 𝐴):(ω ↑_𝑜 𝐴)–1-1-onto→𝑆)
10		f1of 6050	. . . . . . . . 9 ⊢ (^◡(ω CNF 𝐴):(ω ↑_𝑜 𝐴)–1-1-onto→𝑆 → ^◡(ω CNF 𝐴):(ω ↑_𝑜 𝐴)⟶𝑆)
11	8, 9, 10	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡(ω CNF 𝐴):(ω ↑_𝑜 𝐴)⟶𝑆)
12		cnfcom.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (ω ↑_𝑜 𝐴))
13	11, 12	ffvelrnd 6268	. . . . . . 7 ⊢ (𝜑 → (^◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
14	7, 13	syl5eqel 2692	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
15	2, 4, 5, 6, 14	cantnfcl 8447	. . . . 5 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
16	15	simprd 478	. . . 4 ⊢ (𝜑 → dom 𝐺 ∈ ω)
17		elnn 6967	. . . 4 ⊢ ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
18	1, 16, 17	syl2anc 691	. . 3 ⊢ (𝜑 → 𝐼 ∈ ω)
19		eleq1 2676	. . . . . 6 ⊢ (𝑤 = 𝐼 → (𝑤 ∈ dom 𝐺 ↔ 𝐼 ∈ dom 𝐺))
20		suceq 5707	. . . . . . . 8 ⊢ (𝑤 = 𝐼 → suc 𝑤 = suc 𝐼)
21	20	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 = 𝐼 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝐼))
22	20	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 = 𝐼 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝐼))
23		fveq2 6103	. . . . . . . . 9 ⊢ (𝑤 = 𝐼 → (𝐺‘𝑤) = (𝐺‘𝐼))
24	23	oveq2d 6565	. . . . . . . 8 ⊢ (𝑤 = 𝐼 → (ω ↑_𝑜 (𝐺‘𝑤)) = (ω ↑_𝑜 (𝐺‘𝐼)))
25	23	fveq2d 6107	. . . . . . . 8 ⊢ (𝑤 = 𝐼 → (𝐹‘(𝐺‘𝑤)) = (𝐹‘(𝐺‘𝐼)))
26	24, 25	oveq12d 6567	. . . . . . 7 ⊢ (𝑤 = 𝐼 → ((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤))) = ((ω ↑_𝑜 (𝐺‘𝐼)) ·_𝑜 (𝐹‘(𝐺‘𝐼))))
27	21, 22, 26	f1oeq123d 6046	. . . . . 6 ⊢ (𝑤 = 𝐼 → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤))) ↔ (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝐼)) ·_𝑜 (𝐹‘(𝐺‘𝐼)))))
28	19, 27	imbi12d 333	. . . . 5 ⊢ (𝑤 = 𝐼 → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤)))) ↔ (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝐼)) ·_𝑜 (𝐹‘(𝐺‘𝐼))))))
29	28	imbi2d 329	. . . 4 ⊢ (𝑤 = 𝐼 → ((𝜑 → (𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤))))) ↔ (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝐼)) ·_𝑜 (𝐹‘(𝐺‘𝐼)))))))
30		eleq1 2676	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ∈ dom 𝐺 ↔ ∅ ∈ dom 𝐺))
31		suceq 5707	. . . . . . . 8 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅)
32	31	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑇‘suc 𝑤) = (𝑇‘suc ∅))
33	31	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝐻‘suc 𝑤) = (𝐻‘suc ∅))
34		fveq2 6103	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝐺‘𝑤) = (𝐺‘∅))
35	34	oveq2d 6565	. . . . . . . 8 ⊢ (𝑤 = ∅ → (ω ↑_𝑜 (𝐺‘𝑤)) = (ω ↑_𝑜 (𝐺‘∅)))
36	34	fveq2d 6107	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝐹‘(𝐺‘𝑤)) = (𝐹‘(𝐺‘∅)))
37	35, 36	oveq12d 6567	. . . . . . 7 ⊢ (𝑤 = ∅ → ((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤))) = ((ω ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))))
38	32, 33, 37	f1oeq123d 6046	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤))) ↔ (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅)))))
39	30, 38	imbi12d 333	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤)))) ↔ (∅ ∈ dom 𝐺 → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))))))
40		eleq1 2676	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ dom 𝐺 ↔ 𝑦 ∈ dom 𝐺))
41		suceq 5707	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → suc 𝑤 = suc 𝑦)
42	41	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝑦))
43	41	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝑦))
44		fveq2 6103	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐺‘𝑤) = (𝐺‘𝑦))
45	44	oveq2d 6565	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (ω ↑_𝑜 (𝐺‘𝑤)) = (ω ↑_𝑜 (𝐺‘𝑦)))
46	44	fveq2d 6107	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝐹‘(𝐺‘𝑤)) = (𝐹‘(𝐺‘𝑦)))
47	45, 46	oveq12d 6567	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤))) = ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))
48	42, 43, 47	f1oeq123d 6046	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤))) ↔ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦)))))
49	40, 48	imbi12d 333	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤)))) ↔ (𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))))
50		eleq1 2676	. . . . . 6 ⊢ (𝑤 = suc 𝑦 → (𝑤 ∈ dom 𝐺 ↔ suc 𝑦 ∈ dom 𝐺))
51		suceq 5707	. . . . . . . 8 ⊢ (𝑤 = suc 𝑦 → suc 𝑤 = suc suc 𝑦)
52	51	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 = suc 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc suc 𝑦))
53	51	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 = suc 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc suc 𝑦))
54		fveq2 6103	. . . . . . . . 9 ⊢ (𝑤 = suc 𝑦 → (𝐺‘𝑤) = (𝐺‘suc 𝑦))
55	54	oveq2d 6565	. . . . . . . 8 ⊢ (𝑤 = suc 𝑦 → (ω ↑_𝑜 (𝐺‘𝑤)) = (ω ↑_𝑜 (𝐺‘suc 𝑦)))
56	54	fveq2d 6107	. . . . . . . 8 ⊢ (𝑤 = suc 𝑦 → (𝐹‘(𝐺‘𝑤)) = (𝐹‘(𝐺‘suc 𝑦)))
57	55, 56	oveq12d 6567	. . . . . . 7 ⊢ (𝑤 = suc 𝑦 → ((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤))) = ((ω ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))))
58	52, 53, 57	f1oeq123d 6046	. . . . . 6 ⊢ (𝑤 = suc 𝑦 → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤))) ↔ (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦)))))
59	50, 58	imbi12d 333	. . . . 5 ⊢ (𝑤 = suc 𝑦 → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤)))) ↔ (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))))))
60	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐴 ∈ On)
61	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐵 ∈ (ω ↑_𝑜 𝐴))
62		cnfcom.h	. . . . . . 7 ⊢ 𝐻 = seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +_𝑜 𝑧)), ∅)
63		cnfcom.t	. . . . . . 7 ⊢ 𝑇 = seq_𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
64		cnfcom.m	. . . . . . 7 ⊢ 𝑀 = ((ω ↑_𝑜 (𝐺‘𝑘)) ·_𝑜 (𝐹‘(𝐺‘𝑘)))
65		cnfcom.k	. . . . . . 7 ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +_𝑜 𝑥)) ∪ ^◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +_𝑜 𝑥)))
66		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ dom 𝐺)
67	3	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ω ∈ On)
68		suppssdm 7195	. . . . . . . . . . 11 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹
69	2, 4, 5	cantnfs 8446	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
70	14, 69	mpbid 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
71	70	simpld 474	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶ω)
72		fdm 5964	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
73	71, 72	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝐴)
74	68, 73	syl5sseq 3616	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
75		onss 6882	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
76	5, 75	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ On)
77	74, 76	sstrd 3578	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ On)
78	6	oif 8318	. . . . . . . . . 10 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
79	78	ffvelrni 6266	. . . . . . . . 9 ⊢ (∅ ∈ dom 𝐺 → (𝐺‘∅) ∈ (𝐹 supp ∅))
80		ssel2 3563	. . . . . . . . 9 ⊢ (((𝐹 supp ∅) ⊆ On ∧ (𝐺‘∅) ∈ (𝐹 supp ∅)) → (𝐺‘∅) ∈ On)
81	77, 79, 80	syl2an 493	. . . . . . . 8 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ On)
82		peano1 6977	. . . . . . . . 9 ⊢ ∅ ∈ ω
83	82	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ ω)
84		oen0 7553	. . . . . . . 8 ⊢ (((ω ∈ On ∧ (𝐺‘∅) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑_𝑜 (𝐺‘∅)))
85	67, 81, 83, 84	syl21anc 1317	. . . . . . 7 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ (ω ↑_𝑜 (𝐺‘∅)))
86		0ex 4718	. . . . . . . . 9 ⊢ ∅ ∈ V
87	63	seqom0g 7438	. . . . . . . . 9 ⊢ (∅ ∈ V → (𝑇‘∅) = ∅)
88	86, 87	ax-mp 5	. . . . . . . 8 ⊢ (𝑇‘∅) = ∅
89		f1o0 6085	. . . . . . . . . 10 ⊢ ∅:∅–1-1-onto→∅
90	62	seqom0g 7438	. . . . . . . . . . 11 ⊢ (∅ ∈ V → (𝐻‘∅) = ∅)
91		f1oeq2 6041	. . . . . . . . . . 11 ⊢ ((𝐻‘∅) = ∅ → (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
92	86, 90, 91	mp2b 10	. . . . . . . . . 10 ⊢ (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)
93	89, 92	mpbir 220	. . . . . . . . 9 ⊢ ∅:(𝐻‘∅)–1-1-onto→∅
94		f1oeq1 6040	. . . . . . . . 9 ⊢ ((𝑇‘∅) = ∅ → ((𝑇‘∅):(𝐻‘∅)–1-1-onto→∅ ↔ ∅:(𝐻‘∅)–1-1-onto→∅))
95	93, 94	mpbiri 247	. . . . . . . 8 ⊢ ((𝑇‘∅) = ∅ → (𝑇‘∅):(𝐻‘∅)–1-1-onto→∅)
96	88, 95	mp1i 13	. . . . . . 7 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝑇‘∅):(𝐻‘∅)–1-1-onto→∅)
97	2, 60, 61, 7, 6, 62, 63, 64, 65, 66, 85, 96	cnfcomlem 8479	. . . . . 6 ⊢ ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅))))
98	97	ex 449	. . . . 5 ⊢ (𝜑 → (∅ ∈ dom 𝐺 → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑_𝑜 (𝐺‘∅)) ·_𝑜 (𝐹‘(𝐺‘∅)))))
99	6	oicl 8317	. . . . . . . . . 10 ⊢ Ord dom 𝐺
100		ordtr 5654	. . . . . . . . . 10 ⊢ (Ord dom 𝐺 → Tr dom 𝐺)
101	99, 100	ax-mp 5	. . . . . . . . 9 ⊢ Tr dom 𝐺
102		trsuc 5727	. . . . . . . . 9 ⊢ ((Tr dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺) → 𝑦 ∈ dom 𝐺)
103	101, 102	mpan 702	. . . . . . . 8 ⊢ (suc 𝑦 ∈ dom 𝐺 → 𝑦 ∈ dom 𝐺)
104	103	imim1i 61	. . . . . . 7 ⊢ ((𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦)))))
105	5	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → 𝐴 ∈ On)
106	12	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → 𝐵 ∈ (ω ↑_𝑜 𝐴))
107		simprl 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → suc 𝑦 ∈ dom 𝐺)
108	76	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → 𝐴 ⊆ On)
109	74	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝐹 supp ∅) ⊆ 𝐴)
110	78	ffvelrni 6266	. . . . . . . . . . . . . . . . 17 ⊢ (suc 𝑦 ∈ dom 𝐺 → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
111	110	ad2antrl 760	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
112	109, 111	sseldd 3569	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝐺‘suc 𝑦) ∈ 𝐴)
113	108, 112	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝐺‘suc 𝑦) ∈ On)
114		eloni 5650	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘suc 𝑦) ∈ On → Ord (𝐺‘suc 𝑦))
115	113, 114	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → Ord (𝐺‘suc 𝑦))
116		vex 3176	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
117	116	sucid 5721	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ suc 𝑦
118	5, 74	ssexd 4733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V)
119	15	simpld 474	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → E We (𝐹 supp ∅))
120	6	oiiso 8325	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
121	118, 119, 120	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
122	121	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
123	103	ad2antrl 760	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → 𝑦 ∈ dom 𝐺)
124		isorel 6476	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺)) → (𝑦 E suc 𝑦 ↔ (𝐺‘𝑦) E (𝐺‘suc 𝑦)))
125	122, 123, 107, 124	syl12anc 1316	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝑦 E suc 𝑦 ↔ (𝐺‘𝑦) E (𝐺‘suc 𝑦)))
126	116	sucex 6903	. . . . . . . . . . . . . . . 16 ⊢ suc 𝑦 ∈ V
127	126	epelc 4951	. . . . . . . . . . . . . . 15 ⊢ (𝑦 E suc 𝑦 ↔ 𝑦 ∈ suc 𝑦)
128		fvex 6113	. . . . . . . . . . . . . . . 16 ⊢ (𝐺‘suc 𝑦) ∈ V
129	128	epelc 4951	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑦) E (𝐺‘suc 𝑦) ↔ (𝐺‘𝑦) ∈ (𝐺‘suc 𝑦))
130	125, 127, 129	3bitr3g 301	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝑦 ∈ suc 𝑦 ↔ (𝐺‘𝑦) ∈ (𝐺‘suc 𝑦)))
131	117, 130	mpbii 222	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝐺‘𝑦) ∈ (𝐺‘suc 𝑦))
132		ordsucss 6910	. . . . . . . . . . . . 13 ⊢ (Ord (𝐺‘suc 𝑦) → ((𝐺‘𝑦) ∈ (𝐺‘suc 𝑦) → suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)))
133	115, 131, 132	sylc 63	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦))
134	78	ffvelrni 6266	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ dom 𝐺 → (𝐺‘𝑦) ∈ (𝐹 supp ∅))
135	123, 134	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝐺‘𝑦) ∈ (𝐹 supp ∅))
136	109, 135	sseldd 3569	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝐺‘𝑦) ∈ 𝐴)
137	108, 136	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝐺‘𝑦) ∈ On)
138		suceloni 6905	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝑦) ∈ On → suc (𝐺‘𝑦) ∈ On)
139	137, 138	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → suc (𝐺‘𝑦) ∈ On)
140	3	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → ω ∈ On)
141	82	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → ∅ ∈ ω)
142		oewordi 7558	. . . . . . . . . . . . 13 ⊢ (((suc (𝐺‘𝑦) ∈ On ∧ (𝐺‘suc 𝑦) ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → (suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → (ω ↑_𝑜 suc (𝐺‘𝑦)) ⊆ (ω ↑_𝑜 (𝐺‘suc 𝑦))))
143	139, 113, 140, 141, 142	syl31anc 1321	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (suc (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → (ω ↑_𝑜 suc (𝐺‘𝑦)) ⊆ (ω ↑_𝑜 (𝐺‘suc 𝑦))))
144	133, 143	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (ω ↑_𝑜 suc (𝐺‘𝑦)) ⊆ (ω ↑_𝑜 (𝐺‘suc 𝑦)))
145	71	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → 𝐹:𝐴⟶ω)
146	145, 136	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝐹‘(𝐺‘𝑦)) ∈ ω)
147		nnon 6963	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘(𝐺‘𝑦)) ∈ ω → (𝐹‘(𝐺‘𝑦)) ∈ On)
148	146, 147	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝐹‘(𝐺‘𝑦)) ∈ On)
149		oecl 7504	. . . . . . . . . . . . . . 15 ⊢ ((ω ∈ On ∧ (𝐺‘𝑦) ∈ On) → (ω ↑_𝑜 (𝐺‘𝑦)) ∈ On)
150	140, 137, 149	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (ω ↑_𝑜 (𝐺‘𝑦)) ∈ On)
151		oen0 7553	. . . . . . . . . . . . . . 15 ⊢ (((ω ∈ On ∧ (𝐺‘𝑦) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑_𝑜 (𝐺‘𝑦)))
152	140, 137, 141, 151	syl21anc 1317	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → ∅ ∈ (ω ↑_𝑜 (𝐺‘𝑦)))
153		omord2 7534	. . . . . . . . . . . . . 14 ⊢ ((((𝐹‘(𝐺‘𝑦)) ∈ On ∧ ω ∈ On ∧ (ω ↑_𝑜 (𝐺‘𝑦)) ∈ On) ∧ ∅ ∈ (ω ↑_𝑜 (𝐺‘𝑦))) → ((𝐹‘(𝐺‘𝑦)) ∈ ω ↔ ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))) ∈ ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 ω)))
154	148, 140, 150, 152, 153	syl31anc 1321	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → ((𝐹‘(𝐺‘𝑦)) ∈ ω ↔ ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))) ∈ ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 ω)))
155	146, 154	mpbid 221	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))) ∈ ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 ω))
156		oesuc 7494	. . . . . . . . . . . . 13 ⊢ ((ω ∈ On ∧ (𝐺‘𝑦) ∈ On) → (ω ↑_𝑜 suc (𝐺‘𝑦)) = ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 ω))
157	140, 137, 156	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (ω ↑_𝑜 suc (𝐺‘𝑦)) = ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 ω))
158	155, 157	eleqtrrd 2691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))) ∈ (ω ↑_𝑜 suc (𝐺‘𝑦)))
159	144, 158	sseldd 3569	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → ((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))) ∈ (ω ↑_𝑜 (𝐺‘suc 𝑦)))
160		simprr 792	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))
161	2, 105, 106, 7, 6, 62, 63, 64, 65, 107, 159, 160	cnfcomlem 8479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))))) → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))))
162	161	exp32 629	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (suc 𝑦 ∈ dom 𝐺 → ((𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦))) → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))))))
163	162	a2d 29	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))))))
164	104, 163	syl5 33	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → ((𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦))))))
165	164	expcom 450	. . . . 5 ⊢ (𝑦 ∈ ω → (𝜑 → ((𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑦)) ·_𝑜 (𝐹‘(𝐺‘𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑_𝑜 (𝐺‘suc 𝑦)) ·_𝑜 (𝐹‘(𝐺‘suc 𝑦)))))))
166	39, 49, 59, 98, 165	finds2 6986	. . . 4 ⊢ (𝑤 ∈ ω → (𝜑 → (𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝑤)) ·_𝑜 (𝐹‘(𝐺‘𝑤))))))
167	29, 166	vtoclga 3245	. . 3 ⊢ (𝐼 ∈ ω → (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝐼)) ·_𝑜 (𝐹‘(𝐺‘𝐼))))))
168	18, 167	mpcom 37	. 2 ⊢ (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝐼)) ·_𝑜 (𝐹‘(𝐺‘𝐼)))))
169	1, 168	mpd 15	1 ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑_𝑜 (𝐺‘𝐼)) ·_𝑜 (𝐹‘(𝐺‘𝐼))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ↦ cmpt 4643 Tr wtr 4680 E cep 4947 We wwe 4996 ^◡ccnv 5037 dom cdm 5038 Ord word 5639 Oncon0 5640 suc csuc 5642 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 supp csupp 7182 seq_𝜔cseqom 7429 +_𝑜 coa 7444 ·_𝑜 comu 7445 ↑_𝑜 coe 7446 finSupp cfsupp 8158 OrdIsocoi 8297 CNF ccnf 8441

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 ax-inf2 8421

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-oexp 7453 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-cnf 8442

This theorem is referenced by: cnfcom2 8482

W3C validator