ackbij2 - Metamath Proof Explorer

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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 6891	. . . . . 6 ⊢ (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
2		fvex 6904	. . . . . 6 ⊢ (rec(𝐺, ∅)‘𝑎) ∈ V
3	1, 2	f1iun 7929	. . . . 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 6832	. . . . . . . 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 7864	. . . . . . . 8 ⊢ Ord ω
10		r1fin 9767	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (𝑅₁‘𝑎) ∈ Fin)
11		ficardom 9955	. . . . . . . . 9 ⊢ ((𝑅₁‘𝑎) ∈ Fin → (card‘(𝑅₁‘𝑎)) ∈ ω)
12	10, 11	syl 17	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝑎 ∈ ω → (card‘(𝑅₁‘𝑎)) ⊆ ω)
15		f1ss 6793	. . . . . . 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 7862	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → Ord 𝑎)
18		nnord 7862	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → Ord 𝑏)
19		ordtri2or2 6463	. . . . . . . . 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 10236	. . . . . . . . . . 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 10236	. . . . . . . . . 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 3146	. . . . . 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 3067	. . . 4 ⊢ ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎):∪ 𝑎 ∈ ω (𝑅₁‘𝑎)–1-1→ω
31		rdgfun 8415	. . . . . 6 ⊢ Fun rec(𝐺, ∅)
32		funiunfv 7246	. . . . . . 7 ⊢ (Fun rec(𝐺, ∅) → ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎) = ∪ (rec(𝐺, ∅) “ ω))
33	32	eqcomd 2738	. . . . . 6 ⊢ (Fun rec(𝐺, ∅) → ∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎))
34		f1eq1 6782	. . . . . 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 9760	. . . . . . 7 ⊢ (Fun 𝑅₁ ∧ Lim dom 𝑅₁)
37	36	simpli 484	. . . . . 6 ⊢ Fun 𝑅₁
38		funiunfv 7246	. . . . . 6 ⊢ (Fun 𝑅₁ → ∪ 𝑎 ∈ ω (𝑅₁‘𝑎) = ∪ (𝑅₁ “ ω))
39		f1eq2 6783	. . . . . 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 6148	. . . 4 ⊢ ran ∪ (rec(𝐺, ∅) “ ω) = ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎
44		eliun 5001	. . . . . 6 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ ∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎)
45		df-rex 3071	. . . . . 6 ⊢ (∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎 ↔ ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
46		funfn 6578	. . . . . . . . . . . 12 ⊢ (Fun rec(𝐺, ∅) ↔ rec(𝐺, ∅) Fn dom rec(𝐺, ∅))
47	31, 46	mpbi 229	. . . . . . . . . . 11 ⊢ rec(𝐺, ∅) Fn dom rec(𝐺, ∅)
48		rdgdmlim 8416	. . . . . . . . . . . 12 ⊢ Lim dom rec(𝐺, ∅)
49		limomss 7859	. . . . . . . . . . . 12 ⊢ (Lim dom rec(𝐺, ∅) → ω ⊆ dom rec(𝐺, ∅))
50	48, 49	ax-mp 5	. . . . . . . . . . 11 ⊢ ω ⊆ dom rec(𝐺, ∅)
51		fvelimab 6964	. . . . . . . . . . 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 10234	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ω → (rec(𝐺, ∅)‘𝑐):(𝑅₁‘𝑐)–1-1-onto→(card‘(𝑅₁‘𝑐)))
54		f1ofo 6840	. . . . . . . . . . . . . 14 ⊢ ((rec(𝐺, ∅)‘𝑐):(𝑅₁‘𝑐)–1-1-onto→(card‘(𝑅₁‘𝑐)) → (rec(𝐺, ∅)‘𝑐):(𝑅₁‘𝑐)–onto→(card‘(𝑅₁‘𝑐)))
55		forn 6808	. . . . . . . . . . . . . 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 9767	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ω → (𝑅₁‘𝑐) ∈ Fin)
58		ficardom 9955	. . . . . . . . . . . . . . 15 ⊢ ((𝑅₁‘𝑐) ∈ Fin → (card‘(𝑅₁‘𝑐)) ∈ ω)
59	57, 58	syl 17	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ω → (card‘(𝑅₁‘𝑐)) ⊆ ω)
62	56, 61	eqsstrd 4020	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ω → ran (rec(𝐺, ∅)‘𝑐) ⊆ ω)
63		rneq 5935	. . . . . . . . . . . . 13 ⊢ ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran (rec(𝐺, ∅)‘𝑐) = ran 𝑎)
64	63	sseq1d 4013	. . . . . . . . . . . 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 3148	. . . . . . . . . 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 3982	. . . . . . . 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 7880	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → suc 𝑏 ∈ ω)
71		fnfvima 7234	. . . . . . . . 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 3478	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
74		cardnn 9957	. . . . . . . . . . . 12 ⊢ (suc 𝑏 ∈ ω → (card‘suc 𝑏) = suc 𝑏)
75		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (𝑅₁‘suc 𝑏) ∈ V
76	36	simpri 486	. . . . . . . . . . . . . . . . 17 ⊢ Lim dom 𝑅₁
77		limomss 7859	. . . . . . . . . . . . . . . . 17 ⊢ (Lim dom 𝑅₁ → ω ⊆ dom 𝑅₁)
78	76, 77	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ω ⊆ dom 𝑅₁
79	78	sseli 3978	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom 𝑅₁)
80		onssr1 9825	. . . . . . . . . . . . . . 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 8995	. . . . . . . . . . . . . 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 7860	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ On)
85		onenon 9943	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ On → suc 𝑏 ∈ dom card)
86	84, 85	syl 17	. . . . . . . . . . . . . 14 ⊢ (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom card)
87		r1fin 9767	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑏 ∈ ω → (𝑅₁‘suc 𝑏) ∈ Fin)
88		finnum 9942	. . . . . . . . . . . . . . 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 9971	. . . . . . . . . . . . . 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 4021	. . . . . . . . . . 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 6459	. . . . . . . . . 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 10234	. . . . . . . . . 10 ⊢ (suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏):(𝑅₁‘suc 𝑏)–1-1-onto→(card‘(𝑅₁‘suc 𝑏)))
98		f1ofo 6840	. . . . . . . . . 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 6808	. . . . . . . . . 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 2836	. . . . . . . 8 ⊢ (𝑏 ∈ ω → 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏))
102		fvex 6904	. . . . . . . . 9 ⊢ (rec(𝐺, ∅)‘suc 𝑏) ∈ V
103		eleq1 2821	. . . . . . . . . 10 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω)))
104		rneq 5935	. . . . . . . . . . 11 ⊢ (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → ran 𝑎 = ran (rec(𝐺, ∅)‘suc 𝑏))
105	104	eleq2d 2819	. . . . . . . . . 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 3596	. . . . . . . 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 2729	. . . 4 ⊢ ∪ 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 = ω
112	43, 111	eqtri 2760	. . 3 ⊢ ran ∪ (rec(𝐺, ∅) “ ω) = ω
113		dff1o5 6842	. . 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 6821	. . 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 396 ∨ wo 845 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 dom cdm 5676 ran crn 5677 “ cima 5679 Ord word 6363 Oncon0 6364 Lim wlim 6365 suc csuc 6366 Fun wfun 6537 Fn wfn 6538 –1-1→wf1 6540 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 ωcom 7854 reccrdg 8408 ≼ cdom 8936 Fincfn 8938 𝑅₁cr1 9756 cardccrd 9929

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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-oadd 8469 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-r1 9758 df-rank 9759 df-dju 9895 df-card 9933

This theorem is referenced by: r1om 10238

W3C validator