Theorem ackbij2 10179

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 6842	. . . . . 6 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
2		fvex 6855	. . . . . 6 ⊢ (rec(𝐺, ∅)‘𝑎) ∈ V
3	1, 2	f1iun 7876	. . . . 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 10176	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1-onto→(card‘(𝑅1‘𝑎)))
7		f1of1 6783	. . . . . . . 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 7812	. . . . . . . 8 ⊢ Ord ω
10		r1fin 9709	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (𝑅1‘𝑎) ∈ Fin)
11		ficardom 9897	. . . . . . . . 9 ⊢ ((𝑅1‘𝑎) ∈ Fin → (card‘(𝑅1‘𝑎)) ∈ ω)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (card‘(𝑅1‘𝑎)) ∈ ω)
13		ordelss 6333	. . . . . . . 8 ⊢ ((Ord ω ∧ (card‘(𝑅1‘𝑎)) ∈ ω) → (card‘(𝑅1‘𝑎)) ⊆ ω)
14	9, 12, 13	sylancr 587	. . . . . . 7 ⊢ (𝑎 ∈ ω → (card‘(𝑅1‘𝑎)) ⊆ ω)
15		f1ss 6744	. . . . . . 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 584	. . . . . 6 ⊢ (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1→ω)
17		nnord 7810	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → Ord 𝑎)
18		nnord 7810	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → Ord 𝑏)
19		ordtri2or2 6416	. . . . . . . . 9 ⊢ ((Ord 𝑎 ∧ Ord 𝑏) → (𝑎 ⊆ 𝑏 ∨ 𝑏 ⊆ 𝑎))
20	17, 18, 19	syl2an 596	. . . . . . . 8 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎 ⊆ 𝑏 ∨ 𝑏 ⊆ 𝑎))
21	4, 5	ackbij2lem4 10178	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ 𝑎 ∈ ω) ∧ 𝑎 ⊆ 𝑏) → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏))
22	21	ex 413	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ 𝑎 ∈ ω) → (𝑎 ⊆ 𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
23	22	ancoms 459	. . . . . . . . 9 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎 ⊆ 𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
24	4, 5	ackbij2lem4 10178	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑎) → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))
25	24	ex 413	. . . . . . . . 9 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑏 ⊆ 𝑎 → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
26	23, 25	orim12d 963	. . . . . . . 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 3143	. . . . . 6 ⊢ (𝑎 ∈ ω → ∀𝑏 ∈ ω ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
29	16, 28	jca 512	. . . . 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 3070	. . . 4 ⊢ ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ 𝑎 ∈ ω (𝑅1‘𝑎)–1-1→ω
31		rdgfun 8362	. . . . . 6 ⊢ Fun rec(𝐺, ∅)
32		funiunfv 7195	. . . . . . 7 ⊢ (Fun rec(𝐺, ∅) → ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎) = ∪ (rec(𝐺, ∅) “ ω))
33	32	eqcomd 2742	. . . . . 6 ⊢ (Fun rec(𝐺, ∅) → ∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎))
34		f1eq1 6733	. . . . . 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 9702	. . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
37	36	simpli 484	. . . . . 6 ⊢ Fun 𝑅1
38		funiunfv 7195	. . . . . 6 ⊢ (Fun 𝑅1 → ∪ 𝑎 ∈ ω (𝑅1‘𝑎) = ∪ (𝑅1 “ ω))
39		f1eq2 6734	. . . . . 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 277	. . . 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 230	. . 3 ⊢ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1→ω
43		rnuni 6101	. . . 4 ⊢ ran ∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎
44		eliun 4958	. . . . . 6 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ ∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎)
45		df-rex 3074	. . . . . 6 ⊢ (∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎 ↔ ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
46		funfn 6531	. . . . . . . . . . . 12 ⊢ (Fun rec(𝐺, ∅) ↔ rec(𝐺, ∅) Fn dom rec(𝐺, ∅))
47	31, 46	mpbi 229	. . . . . . . . . . 11 ⊢ rec(𝐺, ∅) Fn dom rec(𝐺, ∅)
48		rdgdmlim 8363	. . . . . . . . . . . 12 ⊢ Lim dom rec(𝐺, ∅)
49		limomss 7807	. . . . . . . . . . . 12 ⊢ (Lim dom rec(𝐺, ∅) → ω ⊆ dom rec(𝐺, ∅))
50	48, 49	ax-mp 5	. . . . . . . . . . 11 ⊢ ω ⊆ dom rec(𝐺, ∅)
51		fvelimab 6914	. . . . . . . . . . 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 690	. . . . . . . . . 10 ⊢ (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ ∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎)
53	4, 5	ackbij2lem2 10176	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ω → (rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–1-1-onto→(card‘(𝑅1‘𝑐)))
54		f1ofo 6791	. . . . . . . . . . . . . 14 ⊢ ((rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–1-1-onto→(card‘(𝑅1‘𝑐)) → (rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–onto→(card‘(𝑅1‘𝑐)))
55		forn 6759	. . . . . . . . . . . . . 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 9709	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ω → (𝑅1‘𝑐) ∈ Fin)
58		ficardom 9897	. . . . . . . . . . . . . . 15 ⊢ ((𝑅1‘𝑐) ∈ Fin → (card‘(𝑅1‘𝑐)) ∈ ω)
59	57, 58	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ω → (card‘(𝑅1‘𝑐)) ∈ ω)
60		ordelss 6333	. . . . . . . . . . . . . 14 ⊢ ((Ord ω ∧ (card‘(𝑅1‘𝑐)) ∈ ω) → (card‘(𝑅1‘𝑐)) ⊆ ω)
61	9, 59, 60	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ω → (card‘(𝑅1‘𝑐)) ⊆ ω)
62	56, 61	eqsstrd 3982	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ω → ran (rec(𝐺, ∅)‘𝑐) ⊆ ω)
63		rneq 5891	. . . . . . . . . . . . 13 ⊢ ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran (rec(𝐺, ∅)‘𝑐) = ran 𝑎)
64	63	sseq1d 3975	. . . . . . . . . . . 12 ⊢ ((rec(𝐺, ∅)‘𝑐) = 𝑎 → (ran (rec(𝐺, ∅)‘𝑐) ⊆ ω ↔ ran 𝑎 ⊆ ω))
65	62, 64	syl5ibcom 244	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ω → ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran 𝑎 ⊆ ω))
66	65	rexlimiv 3145	. . . . . . . . . 10 ⊢ (∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎 → ran 𝑎 ⊆ ω)
67	52, 66	sylbi 216	. . . . . . . . 9 ⊢ (𝑎 ∈ (rec(𝐺, ∅) “ ω) → ran 𝑎 ⊆ ω)
68	67	sselda 3944	. . . . . . . 8 ⊢ ((𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
69	68	exlimiv 1933	. . . . . . 7 ⊢ (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
70		peano2 7827	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → suc 𝑏 ∈ ω)
71		fnfvima 7183	. . . . . . . . 9 ⊢ ((rec(𝐺, ∅) Fn dom rec(𝐺, ∅) ∧ ω ⊆ dom rec(𝐺, ∅) ∧ suc 𝑏 ∈ ω) → (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω))
72	47, 50, 70, 71	mp3an12i 1465	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω))
73		vex 3449	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
74		cardnn 9899	. . . . . . . . . . . 12 ⊢ (suc 𝑏 ∈ ω → (card‘suc 𝑏) = suc 𝑏)
75		fvex 6855	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘suc 𝑏) ∈ V
76	36	simpri 486	. . . . . . . . . . . . . . . . 17 ⊢ Lim dom 𝑅1
77		limomss 7807	. . . . . . . . . . . . . . . . 17 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1)
78	76, 77	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ω ⊆ dom 𝑅1
79	78	sseli 3940	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom 𝑅1)
80		onssr1 9767	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ dom 𝑅1 → suc 𝑏 ⊆ (𝑅1‘suc 𝑏))
81	79, 80	syl 17	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ⊆ (𝑅1‘suc 𝑏))
82		ssdomg 8940	. . . . . . . . . . . . . 14 ⊢ ((𝑅1‘suc 𝑏) ∈ V → (suc 𝑏 ⊆ (𝑅1‘suc 𝑏) → suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
83	75, 81, 82	mpsyl 68	. . . . . . . . . . . . 13 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ≼ (𝑅1‘suc 𝑏))
84		nnon 7808	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ On)
85		onenon 9885	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ On → suc 𝑏 ∈ dom card)
86	84, 85	syl 17	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom card)
87		r1fin 9709	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → (𝑅1‘suc 𝑏) ∈ Fin)
88		finnum 9884	. . . . . . . . . . . . . . 15 ⊢ ((𝑅1‘suc 𝑏) ∈ Fin → (𝑅1‘suc 𝑏) ∈ dom card)
89	87, 88	syl 17	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → (𝑅1‘suc 𝑏) ∈ dom card)
90		carddom2 9913	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑏 ∈ dom card ∧ (𝑅1‘suc 𝑏) ∈ dom card) → ((card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)) ↔ suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
91	86, 89, 90	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (suc 𝑏 ∈ ω → ((card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)) ↔ suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
92	83, 91	mpbird 256	. . . . . . . . . . . 12 ⊢ (suc 𝑏 ∈ ω → (card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)))
93	74, 92	eqsstrrd 3983	. . . . . . . . . . 11 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)))
94	70, 93	syl 17	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)))
95		sucssel 6412	. . . . . . . . . 10 ⊢ (𝑏 ∈ V → (suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)) → 𝑏 ∈ (card‘(𝑅1‘suc 𝑏))))
96	73, 94, 95	mpsyl 68	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → 𝑏 ∈ (card‘(𝑅1‘suc 𝑏)))
97	4, 5	ackbij2lem2 10176	. . . . . . . . . 10 ⊢ (suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)))
98		f1ofo 6791	. . . . . . . . . 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)))
99		forn 6759	. . . . . . . . . 10 ⊢ ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–onto→(card‘(𝑅1‘suc 𝑏)) → ran (rec(𝐺, ∅)‘suc 𝑏) = (card‘(𝑅1‘suc 𝑏)))
100	70, 97, 98, 99	4syl 19	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ran (rec(𝐺, ∅)‘suc 𝑏) = (card‘(𝑅1‘suc 𝑏)))
101	96, 100	eleqtrrd 2841	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏))
102		fvex 6855	. . . . . . . . 9 ⊢ (rec(𝐺, ∅)‘suc 𝑏) ∈ V
103		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω)))
104		rneq 5891	. . . . . . . . . . 11 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → ran 𝑎 = ran (rec(𝐺, ∅)‘suc 𝑏))
105	104	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑏 ∈ ran 𝑎 ↔ 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏)))
106	103, 105	anbi12d 631	. . . . . . . . 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))))
107	102, 106	spcev 3565	. . . . . . . 8 ⊢ (((rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏)) → ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
108	72, 101, 107	syl2anc 584	. . . . . . 7 ⊢ (𝑏 ∈ ω → ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
109	69, 108	impbii 208	. . . . . 6 ⊢ (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) ↔ 𝑏 ∈ ω)
110	44, 45, 109	3bitri 296	. . . . 5 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ 𝑏 ∈ ω)
111	110	eqriv 2733	. . . 4 ⊢ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 = ω
112	43, 111	eqtri 2764	. . 3 ⊢ ran ∪ (rec(𝐺, ∅) “ ω) = ω
113		dff1o5 6793	. . 3 ⊢ (∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω ↔ (∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1→ω ∧ ran ∪ (rec(𝐺, ∅) “ ω) = ω))
114	42, 112, 113	mpbir2an 709	. 2 ⊢ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω
115		ackbij.h	. . 3 ⊢ 𝐻 = ∪ (rec(𝐺, ∅) “ ω)
116		f1oeq1 6772	. . 3 ⊢ (𝐻 = ∪ (rec(𝐺, ∅) “ ω) → (𝐻:∪ (𝑅1 “ ω)–1-1-onto→ω ↔ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω))
117	115, 116	ax-mp 5	. 2 ⊢ (𝐻:∪ (𝑅1 “ ω)–1-1-onto→ω ↔ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω)
118	114, 117	mpbir 230	1 ⊢ 𝐻:∪ (𝑅1 “ ω)–1-1-onto→ω

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  {csn 4586  ∪ cuni 4865  ∪ ciun 4954   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634   “ cima 5636  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  Fun wfun 6490   Fn wfn 6491  –1-1→wf1 6493  –onto→wfo 6494  –1-1-onto→wf1o 6495  ‘cfv 6496  ωcom 7802  reccrdg 8355   ≼ cdom 8881  Fincfn 8883  𝑅₁cr1 9698  cardccrd 9871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-r1 9700  df-rank 9701  df-dju 9837  df-card 9875

This theorem is referenced by:  r1om  10180