ackbij2 - Metamath Proof Explorer

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Theorem ackbij2 10238

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 6892	. . . . . 6 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
2		fvex 6905	. . . . . 6 ⊢ (rec(𝐺, ∅)‘𝑎) ∈ V
3	1, 2	f1iun 7930	. . . . 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 10235	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅₁‘𝑎)–1-1-onto→(card‘(𝑅₁‘𝑎)))
7		f1of1 6833	. . . . . . . 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 7865	. . . . . . . 8 ⊢ Ord ω
10		r1fin 9768	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (𝑅₁‘𝑎) ∈ Fin)
11		ficardom 9956	. . . . . . . . 9 ⊢ ((𝑅₁‘𝑎) ∈ Fin → (card‘(𝑅₁‘𝑎)) ∈ ω)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (card‘(𝑅₁‘𝑎)) ∈ ω)
13		ordelss 6381	. . . . . . . 8 ⊢ ((Ord ω ∧ (card‘(𝑅₁‘𝑎)) ∈ ω) → (card‘(𝑅₁‘𝑎)) ⊆ ω)
14	9, 12, 13	sylancr 588	. . . . . . 7 ⊢ (𝑎 ∈ ω → (card‘(𝑅₁‘𝑎)) ⊆ ω)
15		f1ss 6794	. . . . . . 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 585	. . . . . 6 ⊢ (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅₁‘𝑎)–1-1→ω)
17		nnord 7863	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → Ord 𝑎)
18		nnord 7863	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → Ord 𝑏)
19		ordtri2or2 6464	. . . . . . . . 9 ⊢ ((Ord 𝑎 ∧ Ord 𝑏) → (𝑎 ⊆ 𝑏 ∨ 𝑏 ⊆ 𝑎))
20	17, 18, 19	syl2an 597	. . . . . . . 8 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎 ⊆ 𝑏 ∨ 𝑏 ⊆ 𝑎))
21	4, 5	ackbij2lem4 10237	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ 𝑎 ∈ ω) ∧ 𝑎 ⊆ 𝑏) → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏))
22	21	ex 414	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ω ∧ 𝑎 ∈ ω) → (𝑎 ⊆ 𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
23	22	ancoms 460	. . . . . . . . 9 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎 ⊆ 𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
24	4, 5	ackbij2lem4 10237	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏 ⊆ 𝑎) → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))
25	24	ex 414	. . . . . . . . 9 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑏 ⊆ 𝑎 → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
26	23, 25	orim12d 964	. . . . . . . 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 3147	. . . . . 6 ⊢ (𝑎 ∈ ω → ∀𝑏 ∈ ω ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
29	16, 28	jca 513	. . . . 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 3068	. . . 4 ⊢ ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ 𝑎 ∈ ω (𝑅₁‘𝑎)–1-1→ω
31		rdgfun 8416	. . . . . 6 ⊢ Fun rec(𝐺, ∅)
32		funiunfv 7247	. . . . . . 7 ⊢ (Fun rec(𝐺, ∅) → ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎) = ∪ (rec(𝐺, ∅) “ ω))
33	32	eqcomd 2739	. . . . . 6 ⊢ (Fun rec(𝐺, ∅) → ∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎))
34		f1eq1 6783	. . . . . 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 9761	. . . . . . 7 ⊢ (Fun 𝑅₁ ∧ Lim dom 𝑅₁)
37	36	simpli 485	. . . . . 6 ⊢ Fun 𝑅₁
38		funiunfv 7247	. . . . . 6 ⊢ (Fun 𝑅₁ → ∪ 𝑎 ∈ ω (𝑅₁‘𝑎) = ∪ (𝑅₁ “ ω))
39		f1eq2 6784	. . . . . 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 278	. . . 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 6149	. . . 4 ⊢ ran ∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎
44		eliun 5002	. . . . . 6 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ ∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎)
45		df-rex 3072	. . . . . 6 ⊢ (∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎 ↔ ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
46		funfn 6579	. . . . . . . . . . . 12 ⊢ (Fun rec(𝐺, ∅) ↔ rec(𝐺, ∅) Fn dom rec(𝐺, ∅))
47	31, 46	mpbi 229	. . . . . . . . . . 11 ⊢ rec(𝐺, ∅) Fn dom rec(𝐺, ∅)
48		rdgdmlim 8417	. . . . . . . . . . . 12 ⊢ Lim dom rec(𝐺, ∅)
49		limomss 7860	. . . . . . . . . . . 12 ⊢ (Lim dom rec(𝐺, ∅) → ω ⊆ dom rec(𝐺, ∅))
50	48, 49	ax-mp 5	. . . . . . . . . . 11 ⊢ ω ⊆ dom rec(𝐺, ∅)
51		fvelimab 6965	. . . . . . . . . . 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 691	. . . . . . . . . 10 ⊢ (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ ∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎)
53	4, 5	ackbij2lem2 10235	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ω → (rec(𝐺, ∅)‘𝑐):(𝑅₁‘𝑐)–1-1-onto→(card‘(𝑅₁‘𝑐)))
54		f1ofo 6841	. . . . . . . . . . . . . 14 ⊢ ((rec(𝐺, ∅)‘𝑐):(𝑅₁‘𝑐)–1-1-onto→(card‘(𝑅₁‘𝑐)) → (rec(𝐺, ∅)‘𝑐):(𝑅₁‘𝑐)–onto→(card‘(𝑅₁‘𝑐)))
55		forn 6809	. . . . . . . . . . . . . 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 9768	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ω → (𝑅₁‘𝑐) ∈ Fin)
58		ficardom 9956	. . . . . . . . . . . . . . 15 ⊢ ((𝑅₁‘𝑐) ∈ Fin → (card‘(𝑅₁‘𝑐)) ∈ ω)
59	57, 58	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ω → (card‘(𝑅₁‘𝑐)) ∈ ω)
60		ordelss 6381	. . . . . . . . . . . . . 14 ⊢ ((Ord ω ∧ (card‘(𝑅₁‘𝑐)) ∈ ω) → (card‘(𝑅₁‘𝑐)) ⊆ ω)
61	9, 59, 60	sylancr 588	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ω → (card‘(𝑅₁‘𝑐)) ⊆ ω)
62	56, 61	eqsstrd 4021	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ω → ran (rec(𝐺, ∅)‘𝑐) ⊆ ω)
63		rneq 5936	. . . . . . . . . . . . 13 ⊢ ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran (rec(𝐺, ∅)‘𝑐) = ran 𝑎)
64	63	sseq1d 4014	. . . . . . . . . . . 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 3149	. . . . . . . . . 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 3983	. . . . . . . 8 ⊢ ((𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
69	68	exlimiv 1934	. . . . . . 7 ⊢ (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
70		peano2 7881	. . . . . . . . 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 1466	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω))
73		vex 3479	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
74		cardnn 9958	. . . . . . . . . . . 12 ⊢ (suc 𝑏 ∈ ω → (card‘suc 𝑏) = suc 𝑏)
75		fvex 6905	. . . . . . . . . . . . . 14 ⊢ (𝑅₁‘suc 𝑏) ∈ V
76	36	simpri 487	. . . . . . . . . . . . . . . . 17 ⊢ Lim dom 𝑅₁
77		limomss 7860	. . . . . . . . . . . . . . . . 17 ⊢ (Lim dom 𝑅₁ → ω ⊆ dom 𝑅₁)
78	76, 77	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ω ⊆ dom 𝑅₁
79	78	sseli 3979	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom 𝑅₁)
80		onssr1 9826	. . . . . . . . . . . . . . 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 8996	. . . . . . . . . . . . . 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 7861	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ On)
85		onenon 9944	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ On → suc 𝑏 ∈ dom card)
86	84, 85	syl 17	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom card)
87		r1fin 9768	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → (𝑅₁‘suc 𝑏) ∈ Fin)
88		finnum 9943	. . . . . . . . . . . . . . 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 9972	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (suc 𝑏 ∈ ω → ((card‘suc 𝑏) ⊆ (card‘(𝑅₁‘suc 𝑏)) ↔ suc 𝑏 ≼ (𝑅₁‘suc 𝑏)))
92	83, 91	mpbird 257	. . . . . . . . . . . 12 ⊢ (suc 𝑏 ∈ ω → (card‘suc 𝑏) ⊆ (card‘(𝑅₁‘suc 𝑏)))
93	74, 92	eqsstrrd 4022	. . . . . . . . . . 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 6460	. . . . . . . . . 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 10235	. . . . . . . . . 10 ⊢ (suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏):(𝑅₁‘suc 𝑏)–1-1-onto→(card‘(𝑅₁‘suc 𝑏)))
98		f1ofo 6841	. . . . . . . . . 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 6809	. . . . . . . . . 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 2837	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏))
102		fvex 6905	. . . . . . . . 9 ⊢ (rec(𝐺, ∅)‘suc 𝑏) ∈ V
103		eleq1 2822	. . . . . . . . . 10 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω)))
104		rneq 5936	. . . . . . . . . . 11 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → ran 𝑎 = ran (rec(𝐺, ∅)‘suc 𝑏))
105	104	eleq2d 2820	. . . . . . . . . 10 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑏 ∈ ran 𝑎 ↔ 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏)))
106	103, 105	anbi12d 632	. . . . . . . . 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 3597	. . . . . . . 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 585	. . . . . . 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 297	. . . . 5 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ 𝑏 ∈ ω)
111	110	eqriv 2730	. . . 4 ⊢ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 = ω
112	43, 111	eqtri 2761	. . 3 ⊢ ran ∪ (rec(𝐺, ∅) “ ω) = ω
113		dff1o5 6843	. . 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 710	. 2 ⊢ ∪ (rec(𝐺, ∅) “ ω):∪ (𝑅₁ “ ω)–1-1-onto→ω
115		ackbij.h	. . 3 ⊢ 𝐻 = ∪ (rec(𝐺, ∅) “ ω)
116		f1oeq1 6822	. . 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

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 {csn 4629 ∪ cuni 4909 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 dom cdm 5677 ran crn 5678 “ cima 5680 Ord word 6364 Oncon0 6365 Lim wlim 6366 suc csuc 6367 Fun wfun 6538 Fn wfn 6539 –1-1→wf1 6541 –onto→wfo 6542 –1-1-onto→wf1o 6543 ‘cfv 6544 ωcom 7855 reccrdg 8409 ≼ cdom 8937 Fincfn 8939 𝑅₁cr1 9757 cardccrd 9930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-r1 9759 df-rank 9760 df-dju 9896 df-card 9934

This theorem is referenced by: r1om 10239

W3C validator