Theorem ackbij2 10237

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 6885	. . . . . 6 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
2		fvex 6898	. . . . . 6 ⊢ (rec(𝐺, ∅)‘𝑎) ∈ V
3	1, 2	f1iun 7943	. . . . 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 10234	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1-onto→(card‘(𝑅1‘𝑎)))
7		f1of1 6823	. . . . . . . 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 18	. . . . . . 7 ⊢ (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1→(card‘(𝑅1‘𝑎)))
9		ordom 7874	. . . . . . . 8 ⊢ Ord ω
10		r1fin 9748	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (𝑅1‘𝑎) ∈ Fin)
11		ficardom 9959	. . . . . . . . 9 ⊢ ((𝑅1‘𝑎) ∈ Fin → (card‘(𝑅1‘𝑎)) ∈ ω)
12	10, 11	syl 18	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (card‘(𝑅1‘𝑎)) ∈ ω)
13		ordelss 6380	. . . . . . . 8 ⊢ ((Ord ω ∧ (card‘(𝑅1‘𝑎)) ∈ ω) → (card‘(𝑅1‘𝑎)) ⊆ ω)
14	9, 12, 13	sylancr 599	. . . . . . 7 ⊢ (𝑎 ∈ ω → (card‘(𝑅1‘𝑎)) ⊆ ω)
15		f1ss 6785	. . . . . . 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 596	. . . . . 6 ⊢ (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1‘𝑎)–1-1→ω)
17		nnord 7872	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → Ord 𝑎)
18		nnord 7872	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → Ord 𝑏)
19		ordtri2or2 6466	. . . . . . . . 9 ⊢ ((Ord 𝑎 ∧ Ord 𝑏) → (𝑎 ⊆ 𝑏 ∨ 𝑏 ⊆ 𝑎))
20	17, 18, 19	syl2an 608	. . . . . . . 8 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎 ⊆ 𝑏 ∨ 𝑏 ⊆ 𝑎))
21	4, 5	ackbij2lem4 10236	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ 𝑎 ∈ ω) ∧ 𝑎 ⊆ 𝑏) → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏))
22	21	ex 418	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ 𝑎 ∈ ω) → (𝑎 ⊆ 𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
23	22	ancoms 464	. . . . . . . . 9 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎 ⊆ 𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
24	4, 5	ackbij2lem4 10236	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑎) → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))
25	24	ex 418	. . . . . . . . 9 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑏 ⊆ 𝑎 → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
26	23, 25	orim12d 979	. . . . . . . 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 16	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
28	27	ralrimiva 3159	. . . . . 6 ⊢ (𝑎 ∈ ω → ∀𝑏 ∈ ω ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
29	16, 28	jca 521	. . . . 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 3087	. . . 4 ⊢ ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ 𝑎 ∈ ω (𝑅1‘𝑎)–1-1→ω
31		rdgfun 8405	. . . . . 6 ⊢ Fun rec(𝐺, ∅)
32		funiunfv 7248	. . . . . . 7 ⊢ (Fun rec(𝐺, ∅) → ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎) = ∪ (rec(𝐺, ∅) “ ω))
33	32	eqcomd 2771	. . . . . 6 ⊢ (Fun rec(𝐺, ∅) → ∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎))
34		f1eq1 6773	. . . . . 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 9741	. . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
37	36	simpli 489	. . . . . 6 ⊢ Fun 𝑅1
38		funiunfv 7248	. . . . . 6 ⊢ (Fun 𝑅1 → ∪ 𝑎 ∈ ω (𝑅1‘𝑎) = ∪ (𝑅1 “ ω))
39		f1eq2 6774	. . . . . 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 281	. . . 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 234	. . 3 ⊢ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1→ω
43		rnuni 6148	. . . 4 ⊢ ran ∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎
44		eliun 4962	. . . . . 6 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ ∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎)
45		df-rex 3092	. . . . . 6 ⊢ (∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎 ↔ ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
46		funfn 6570	. . . . . . . . . . . 12 ⊢ (Fun rec(𝐺, ∅) ↔ rec(𝐺, ∅) Fn dom rec(𝐺, ∅))
47	31, 46	mpbi 233	. . . . . . . . . . 11 ⊢ rec(𝐺, ∅) Fn dom rec(𝐺, ∅)
48		rdgdmlim 8406	. . . . . . . . . . . 12 ⊢ Lim dom rec(𝐺, ∅)
49		limomss 7869	. . . . . . . . . . . 12 ⊢ (Lim dom rec(𝐺, ∅) → ω ⊆ dom rec(𝐺, ∅))
50	48, 49	ax-mp 5	. . . . . . . . . . 11 ⊢ ω ⊆ dom rec(𝐺, ∅)
51		fvelimab 6957	. . . . . . . . . . 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 705	. . . . . . . . . 10 ⊢ (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ ∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎)
53	4, 5	ackbij2lem2 10234	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ω → (rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–1-1-onto→(card‘(𝑅1‘𝑐)))
54		f1ofo 6832	. . . . . . . . . . . . . 14 ⊢ ((rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–1-1-onto→(card‘(𝑅1‘𝑐)) → (rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–onto→(card‘(𝑅1‘𝑐)))
55		forn 6799	. . . . . . . . . . . . . 14 ⊢ ((rec(𝐺, ∅)‘𝑐):(𝑅1‘𝑐)–onto→(card‘(𝑅1‘𝑐)) → ran (rec(𝐺, ∅)‘𝑐) = (card‘(𝑅1‘𝑐)))
56	53, 54, 55	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ω → ran (rec(𝐺, ∅)‘𝑐) = (card‘(𝑅1‘𝑐)))
57		r1fin 9748	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ω → (𝑅1‘𝑐) ∈ Fin)
58		ficardom 9959	. . . . . . . . . . . . . . 15 ⊢ ((𝑅1‘𝑐) ∈ Fin → (card‘(𝑅1‘𝑐)) ∈ ω)
59	57, 58	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ω → (card‘(𝑅1‘𝑐)) ∈ ω)
60		ordelss 6380	. . . . . . . . . . . . . 14 ⊢ ((Ord ω ∧ (card‘(𝑅1‘𝑐)) ∈ ω) → (card‘(𝑅1‘𝑐)) ⊆ ω)
61	9, 59, 60	sylancr 599	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ω → (card‘(𝑅1‘𝑐)) ⊆ ω)
62	56, 61	eqsstrd 3972	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ω → ran (rec(𝐺, ∅)‘𝑐) ⊆ ω)
63		rneq 5928	. . . . . . . . . . . . 13 ⊢ ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran (rec(𝐺, ∅)‘𝑐) = ran 𝑎)
64	63	sseq1d 3969	. . . . . . . . . . . 12 ⊢ ((rec(𝐺, ∅)‘𝑐) = 𝑎 → (ran (rec(𝐺, ∅)‘𝑐) ⊆ ω ↔ ran 𝑎 ⊆ ω))
65	62, 64	syl5ibcom 248	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ω → ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran 𝑎 ⊆ ω))
66	65	rexlimiv 3161	. . . . . . . . . 10 ⊢ (∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎 → ran 𝑎 ⊆ ω)
67	52, 66	sylbi 220	. . . . . . . . 9 ⊢ (𝑎 ∈ (rec(𝐺, ∅) “ ω) → ran 𝑎 ⊆ ω)
68	67	sselda 3938	. . . . . . . 8 ⊢ ((𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
69	68	exlimiv 1963	. . . . . . 7 ⊢ (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
70		peano2 7888	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → suc 𝑏 ∈ ω)
71		fnfvima 7235	. . . . . . . . 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 1494	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω))
73		vex 3461	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
74		cardnn 9961	. . . . . . . . . . . 12 ⊢ (suc 𝑏 ∈ ω → (card‘suc 𝑏) = suc 𝑏)
75		fvex 6898	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘suc 𝑏) ∈ V
76	36	simpri 491	. . . . . . . . . . . . . . . . 17 ⊢ Lim dom 𝑅1
77		limomss 7869	. . . . . . . . . . . . . . . . 17 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1)
78	76, 77	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ω ⊆ dom 𝑅1
79	78	sseli 3934	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom 𝑅1)
80		onssr1 9806	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ dom 𝑅1 → suc 𝑏 ⊆ (𝑅1‘suc 𝑏))
81	79, 80	syl 18	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ⊆ (𝑅1‘suc 𝑏))
82		ssdomg 8999	. . . . . . . . . . . . . 14 ⊢ ((𝑅1‘suc 𝑏) ∈ V → (suc 𝑏 ⊆ (𝑅1‘suc 𝑏) → suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
83	75, 81, 82	mpsyl 69	. . . . . . . . . . . . 13 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ≼ (𝑅1‘suc 𝑏))
84		nnon 7870	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ On)
85		onenon 9947	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ On → suc 𝑏 ∈ dom card)
86	84, 85	syl 18	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom card)
87		r1fin 9748	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → (𝑅1‘suc 𝑏) ∈ Fin)
88		finnum 9946	. . . . . . . . . . . . . . 15 ⊢ ((𝑅1‘suc 𝑏) ∈ Fin → (𝑅1‘suc 𝑏) ∈ dom card)
89	87, 88	syl 18	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → (𝑅1‘suc 𝑏) ∈ dom card)
90		carddom2 9975	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 13 ⊢ (suc 𝑏 ∈ ω → ((card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)) ↔ suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
92	83, 91	mpbird 260	. . . . . . . . . . . 12 ⊢ (suc 𝑏 ∈ ω → (card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)))
93	74, 92	eqsstrrd 3973	. . . . . . . . . . 11 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)))
94	70, 93	syl 18	. . . . . . . . . 10 ⊢ (𝑏 ∈ ω → suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)))
95		sucssel 6462	. . . . . . . . . 10 ⊢ (𝑏 ∈ V → (suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)) → 𝑏 ∈ (card‘(𝑅1‘suc 𝑏))))
96	73, 94, 95	mpsyl 69	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → 𝑏 ∈ (card‘(𝑅1‘suc 𝑏)))
97	4, 5	ackbij2lem2 10234	. . . . . . . . . 10 ⊢ (suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)))
98		f1ofo 6832	. . . . . . . . . 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 6799	. . . . . . . . . 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 20	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ran (rec(𝐺, ∅)‘suc 𝑏) = (card‘(𝑅1‘suc 𝑏)))
101	96, 100	eleqtrrd 2868	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏))
102		fvex 6898	. . . . . . . . 9 ⊢ (rec(𝐺, ∅)‘suc 𝑏) ∈ V
103		eleq1 2853	. . . . . . . . . 10 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω)))
104		rneq 5928	. . . . . . . . . . 11 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → ran 𝑎 = ran (rec(𝐺, ∅)‘suc 𝑏))
105	104	eleq2d 2851	. . . . . . . . . 10 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑏 ∈ ran 𝑎 ↔ 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏)))
106	103, 105	anbi12d 644	. . . . . . . . 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 3567	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝑏 ∈ ω → ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
109	69, 108	impbii 212	. . . . . 6 ⊢ (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) ↔ 𝑏 ∈ ω)
110	44, 45, 109	3bitri 300	. . . . 5 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ 𝑏 ∈ ω)
111	110	eqriv 2762	. . . 4 ⊢ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 = ω
112	43, 111	eqtri 2788	. . 3 ⊢ ran ∪ (rec(𝐺, ∅) “ ω) = ω
113		dff1o5 6834	. . 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 724	. 2 ⊢ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω
115		ackbij.h	. . 3 ⊢ 𝐻 = ∪ (rec(𝐺, ∅) “ ω)
116		f1oeq1 6812	. . 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 234	1 ⊢ 𝐻:∪ (𝑅1 “ ω)–1-1-onto→ω

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {csn 4591  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661  dom cdm 5663  ran crn 5664   “ cima 5666  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  Fun wfun 6534   Fn wfn 6535  –1-1→wf1 6537  –onto→wfo 6538  –1-1-onto→wf1o 6539  ‘cfv 6540  ωcom 7864  reccrdg 8398   ≼ cdom 8943  Fincfn 8945  𝑅₁cr1 9737  cardccrd 9933

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-er 8696  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-r1 9739  df-rank 9740  df-dju 9899  df-card 9937

This theorem is used by:  r1om  10238