Theorem ackbij2 8948

Description: The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Hypotheses

Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
ackbij.g	⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))
ackbij.h	⊢ 𝐻 = ∪ (rec(𝐺, ∅) “ ω)

Assertion

Ref	Expression
ackbij2	⊢ 𝐻:∪ (𝑅1 “ ω)–1-1-onto→ω

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦

Proof of Theorem ackbij2

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6103	. . . . . 6 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
2		fvex 6113	. . . . . 6 ⊢ (rec(𝐺, ∅)‘𝑎) ∈ V
3	1, 2	fun11iun 7019	. . . . 5 ⊢ (∀𝑎 ∈ ω ((rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1→ω ∧ ∀𝑏 ∈ ω ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))) → ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ 𝑎 ∈ ω (𝑅1‘𝑎)–1-1→ω)
4		ackbij.f	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
5		ackbij.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))
6	4, 5	ackbij2lem2 8945	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1-onto→(card‘(𝑅1‘𝑎)))
7		f1of1 6049	. . . . . . . 8 ⊢ ((rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1-onto→(card‘(𝑅1‘𝑎)) → (rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1→(card‘(𝑅1‘𝑎)))
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1→(card‘(𝑅1‘𝑎)))
9		ordom 6966	. . . . . . . 8 ⊢ Ord ω
10		r1fin 8519	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (𝑅1‘𝑎) ∈ Fin)
11		ficardom 8670	. . . . . . . . 9 ⊢ ((𝑅1‘𝑎) ∈ Fin → (card‘(𝑅1‘𝑎)) ∈ ω)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (card‘(𝑅1‘𝑎)) ∈ ω)
13		ordelss 5656	. . . . . . . 8 ⊢ ((Ord ω ∧ (card‘(𝑅1‘𝑎)) ∈ ω) → (card‘(𝑅1‘𝑎)) ⊆ ω)
14	9, 12, 13	sylancr 694	. . . . . . 7 ⊢ (𝑎 ∈ ω → (card‘(𝑅1‘𝑎)) ⊆ ω)
15		f1ss 6019	. . . . . . 7 ⊢ (((rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1→(card‘(𝑅1‘𝑎)) ∧ (card‘(𝑅1‘𝑎)) ⊆ ω) → (rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1→ω)
16	8, 14, 15	syl2anc 691	. . . . . 6 ⊢ (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1→ω)
17		nnord 6965	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → Ord 𝑎)
18		nnord 6965	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → Ord 𝑏)
19		ordtri2or2 5740	. . . . . . . . 9 ⊢ ((Ord 𝑎 ∧ Ord 𝑏) → (𝑎 ⊆ 𝑏 ∨ 𝑏 ⊆ 𝑎))
20	17, 18, 19	syl2an 493	. . . . . . . 8 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎 ⊆ 𝑏 ∨ 𝑏 ⊆ 𝑎))
21	4, 5	ackbij2lem4 8947	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ 𝑎 ∈ ω) ∧ 𝑎 ⊆ 𝑏) → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏))
22	21	ex 449	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ 𝑎 ∈ ω) → (𝑎 ⊆ 𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
23	22	ancoms 468	. . . . . . . . 9 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎 ⊆ 𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
24	4, 5	ackbij2lem4 8947	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑎) → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))
25	24	ex 449	. . . . . . . . 9 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑏 ⊆ 𝑎 → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
26	23, 25	orim12d 879	. . . . . . . 8 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝑎 ⊆ 𝑏 ∨ 𝑏 ⊆ 𝑎) → ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))))
27	20, 26	mpd 15	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
28	27	ralrimiva 2949	. . . . . 6 ⊢ (𝑎 ∈ ω → ∀𝑏 ∈ ω ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
29	16, 28	jca 553	. . . . 5 ⊢ (𝑎 ∈ ω → ((rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1→ω ∧ ∀𝑏 ∈ ω ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))))
30	3, 29	mprg 2910	. . . 4 ⊢ ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ 𝑎 ∈ ω (𝑅1‘𝑎)–1-1→ω
31		rdgfun 7399	. . . . . 6 ⊢ Fun rec(𝐺, ∅)
32		funiunfv 6410	. . . . . . 7 ⊢ (Fun rec(𝐺, ∅) → ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎) = ∪ (rec(𝐺, ∅) “ ω))
33	32	eqcomd 2616	. . . . . 6 ⊢ (Fun rec(𝐺, ∅) → ∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎))
34		f1eq1 6009	. . . . . 6 ⊢ (∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎) → (∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1→ω ↔ ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ (𝑅1 “ ω)–1-1→ω))
35	31, 33, 34	mp2b 10	. . . . 5 ⊢ (∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1→ω ↔ ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ (𝑅1 “ ω)–1-1→ω)
36		r1funlim 8512	. . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
37	36	simpli 473	. . . . . 6 ⊢ Fun 𝑅1
38		funiunfv 6410	. . . . . 6 ⊢ (Fun 𝑅1 → ∪ 𝑎 ∈ ω (𝑅1‘𝑎) = ∪ (𝑅1 “ ω))
39		f1eq2 6010	. . . . . 6 ⊢ (∪ 𝑎 ∈ ω (𝑅1‘𝑎) = ∪ (𝑅1 “ ω) → (∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ 𝑎 ∈ ω (𝑅1‘𝑎)–1-1→ω ↔ ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ (𝑅1 “ ω)–1-1→ω))
40	37, 38, 39	mp2b 10	. . . . 5 ⊢ (∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ 𝑎 ∈ ω (𝑅1‘𝑎)–1-1→ω ↔ ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ (𝑅1 “ ω)–1-1→ω)
41	35, 40	bitr4i 266	. . . 4 ⊢ (∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1→ω ↔ ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ 𝑎 ∈ ω (𝑅1‘𝑎)–1-1→ω)
42	30, 41	mpbir 220	. . 3 ⊢ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1→ω
43		rnuni 5463	. . . 4 ⊢ ran ∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎
44		eliun 4460	. . . . . 6 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ ∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎)
45		df-rex 2902	. . . . . 6 ⊢ (∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎 ↔ ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
46		funfn 5833	. . . . . . . . . . . 12 ⊢ (Fun rec(𝐺, ∅) ↔ rec(𝐺, ∅) Fn dom rec(𝐺, ∅))
47	31, 46	mpbi 219	. . . . . . . . . . 11 ⊢ rec(𝐺, ∅) Fn dom rec(𝐺, ∅)
48		rdgdmlim 7400	. . . . . . . . . . . 12 ⊢ Lim dom rec(𝐺, ∅)
49		limomss 6962	. . . . . . . . . . . 12 ⊢ (Lim dom rec(𝐺, ∅) → ω ⊆ dom rec(𝐺, ∅))
50	48, 49	ax-mp 5	. . . . . . . . . . 11 ⊢ ω ⊆ dom rec(𝐺, ∅)
51		fvelimab 6163	. . . . . . . . . . 11 ⊢ ((rec(𝐺, ∅) Fn dom rec(𝐺, ∅) ∧ ω ⊆ dom rec(𝐺, ∅)) → (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ ∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎))
52	47, 50, 51	mp2an 704	. . . . . . . . . 10 ⊢ (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ ∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎)
53	4, 5	ackbij2lem2 8945	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ω → (rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–1-1-onto→(card‘(𝑅1‘𝑐)))
54		f1ofo 6057	. . . . . . . . . . . . . 14 ⊢ ((rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–1-1-onto→(card‘(𝑅1‘𝑐)) → (rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–onto→(card‘(𝑅1‘𝑐)))
55		forn 6031	. . . . . . . . . . . . . 14 ⊢ ((rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–onto→(card‘(𝑅1‘𝑐)) → ran (rec(𝐺, ∅)‘𝑐) = (card‘(𝑅1‘𝑐)))
56	53, 54, 55	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ω → ran (rec(𝐺, ∅)‘𝑐) = (card‘(𝑅1‘𝑐)))
57		r1fin 8519	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ω → (𝑅1‘𝑐) ∈ Fin)
58		ficardom 8670	. . . . . . . . . . . . . . 15 ⊢ ((𝑅1‘𝑐) ∈ Fin → (card‘(𝑅1‘𝑐)) ∈ ω)
59	57, 58	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ω → (card‘(𝑅1‘𝑐)) ∈ ω)
60		ordelss 5656	. . . . . . . . . . . . . 14 ⊢ ((Ord ω ∧ (card‘(𝑅1‘𝑐)) ∈ ω) → (card‘(𝑅1‘𝑐)) ⊆ ω)
61	9, 59, 60	sylancr 694	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ω → (card‘(𝑅1‘𝑐)) ⊆ ω)
62	56, 61	eqsstrd 3602	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ω → ran (rec(𝐺, ∅)‘𝑐) ⊆ ω)
63		rneq 5272	. . . . . . . . . . . . 13 ⊢ ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran (rec(𝐺, ∅)‘𝑐) = ran 𝑎)
64	63	sseq1d 3595	. . . . . . . . . . . 12 ⊢ ((rec(𝐺, ∅)‘𝑐) = 𝑎 → (ran (rec(𝐺, ∅)‘𝑐) ⊆ ω ↔ ran 𝑎 ⊆ ω))
65	62, 64	syl5ibcom 234	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ω → ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran 𝑎 ⊆ ω))
66	65	rexlimiv 3009	. . . . . . . . . 10 ⊢ (∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎 → ran 𝑎 ⊆ ω)
67	52, 66	sylbi 206	. . . . . . . . 9 ⊢ (𝑎 ∈ (rec(𝐺, ∅) “ ω) → ran 𝑎 ⊆ ω)
68	67	sselda 3568	. . . . . . . 8 ⊢ ((𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
69	68	exlimiv 1845	. . . . . . 7 ⊢ (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
70		peano2 6978	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → suc 𝑏 ∈ ω)
71		fnfvima 6400	. . . . . . . . . 10 ⊢ ((rec(𝐺, ∅) Fn dom rec(𝐺, ∅) ∧ ω ⊆ dom rec(𝐺, ∅) ∧ suc 𝑏 ∈ ω) → (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω))
72	47, 50, 71	mp3an12 1406	. . . . . . . . 9 ⊢ (suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω))
73	70, 72	syl 17	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω))
74		vex 3176	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
75		cardnn 8672	. . . . . . . . . . . 12 ⊢ (suc 𝑏 ∈ ω → (card‘suc 𝑏) = suc 𝑏)
76		fvex 6113	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘suc 𝑏) ∈ V
77	36	simpri 477	. . . . . . . . . . . . . . . . 17 ⊢ Lim dom 𝑅1
78		limomss 6962	. . . . . . . . . . . . . . . . 17 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1)
79	77, 78	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ω ⊆ dom 𝑅1
80	79	sseli 3564	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom 𝑅1)
81		onssr1 8577	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ dom 𝑅1 → suc 𝑏 ⊆ (𝑅1‘suc 𝑏))
82	80, 81	syl 17	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ⊆ (𝑅1‘suc 𝑏))
83		ssdomg 7887	. . . . . . . . . . . . . 14 ⊢ ((𝑅1‘suc 𝑏) ∈ V → (suc 𝑏 ⊆ (𝑅1‘suc 𝑏) → suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
84	76, 82, 83	mpsyl 66	. . . . . . . . . . . . 13 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ≼ (𝑅1‘suc 𝑏))
85		nnon 6963	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ On)
86		onenon 8658	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ On → suc 𝑏 ∈ dom card)
87	85, 86	syl 17	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom card)
88		r1fin 8519	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → (𝑅1‘suc 𝑏) ∈ Fin)
89		finnum 8657	. . . . . . . . . . . . . . 15 ⊢ ((𝑅1‘suc 𝑏) ∈ Fin → (𝑅1‘suc 𝑏) ∈ dom card)
90	88, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → (𝑅1‘suc 𝑏) ∈ dom card)
91		carddom2 8686	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑏 ∈ dom card ∧ (𝑅1‘suc 𝑏) ∈ dom card) → ((card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)) ↔ suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
92	87, 90, 91	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (suc 𝑏 ∈ ω → ((card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)) ↔ suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
93	84, 92	mpbird 246	. . . . . . . . . . . 12 ⊢ (suc 𝑏 ∈ ω → (card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)))
94	75, 93	eqsstr3d 3603	. . . . . . . . . . 11 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)))
95	70, 94	syl 17	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)))
96		sucssel 5736	. . . . . . . . . 10 ⊢ (𝑏 ∈ V → (suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)) → 𝑏 ∈ (card‘(𝑅1‘suc 𝑏))))
97	74, 95, 96	mpsyl 66	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → 𝑏 ∈ (card‘(𝑅1‘suc 𝑏)))
98	4, 5	ackbij2lem2 8945	. . . . . . . . . 10 ⊢ (suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)))
99		f1ofo 6057	. . . . . . . . . 10 ⊢ ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–onto→(card‘(𝑅1‘suc 𝑏)))
100		forn 6031	. . . . . . . . . 10 ⊢ ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–onto→(card‘(𝑅1‘suc 𝑏)) → ran (rec(𝐺, ∅)‘suc 𝑏) = (card‘(𝑅1‘suc 𝑏)))
101	70, 98, 99, 100	4syl 19	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ran (rec(𝐺, ∅)‘suc 𝑏) = (card‘(𝑅1‘suc 𝑏)))
102	97, 101	eleqtrrd 2691	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏))
103		fvex 6113	. . . . . . . . 9 ⊢ (rec(𝐺, ∅)‘suc 𝑏) ∈ V
104		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω)))
105		rneq 5272	. . . . . . . . . . 11 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → ran 𝑎 = ran (rec(𝐺, ∅)‘suc 𝑏))
106	105	eleq2d 2673	. . . . . . . . . 10 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑏 ∈ ran 𝑎 ↔ 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏)))
107	104, 106	anbi12d 743	. . . . . . . . 9 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → ((𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) ↔ ((rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏))))
108	103, 107	spcev 3273	. . . . . . . 8 ⊢ (((rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏)) → ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
109	73, 102, 108	syl2anc 691	. . . . . . 7 ⊢ (𝑏 ∈ ω → ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
110	69, 109	impbii 198	. . . . . 6 ⊢ (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) ↔ 𝑏 ∈ ω)
111	44, 45, 110	3bitri 285	. . . . 5 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ 𝑏 ∈ ω)
112	111	eqriv 2607	. . . 4 ⊢ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 = ω
113	43, 112	eqtri 2632	. . 3 ⊢ ran ∪ (rec(𝐺, ∅) “ ω) = ω
114		dff1o5 6059	. . 3 ⊢ (∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω ↔ (∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1→ω ∧ ran ∪ (rec(𝐺, ∅) “ ω) = ω))
115	42, 113, 114	mpbir2an 957	. 2 ⊢ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω
116		ackbij.h	. . 3 ⊢ 𝐻 = ∪ (rec(𝐺, ∅) “ ω)
117		f1oeq1 6040	. . 3 ⊢ (𝐻 = ∪ (rec(𝐺, ∅) “ ω) → (𝐻:∪ (𝑅1 “ ω)–1-1-onto→ω ↔ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω))
118	116, 117	ax-mp 5	. 2 ⊢ (𝐻:∪ (𝑅1 “ ω)–1-1-onto→ω ↔ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω)
119	115, 118	mpbir 220	1 ⊢ 𝐻:∪ (𝑅1 “ ω)–1-1-onto→ω

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   “ cima 5041  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  ωcom 6957  reccrdg 7392   ≼ cdom 7839  Fincfn 7841  𝑅₁cr1 8508  cardccrd 8644

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-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-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-r1 8510  df-rank 8511  df-card 8648  df-cda 8873

This theorem is referenced by:  r1om  8949