Theorem dnnumch1 37614

Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 8853. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypotheses

Ref	Expression
dnnumch.f	⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
dnnumch.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
dnnumch.g	⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))

Assertion

Ref	Expression
dnnumch1	⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem dnnumch1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dnnumch.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		recsval 7500	. . . . . . 7 ⊢ (𝑥 ∈ On → (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)))
3		dnnumch.f	. . . . . . . 8 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
4	3	fveq1i 6192	. . . . . . 7 ⊢ (𝐹‘𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥)
5	3	tfr1 7493	. . . . . . . . . . 11 ⊢ 𝐹 Fn On
6		fnfun 5988	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → Fun 𝐹)
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 𝐹
8		vex 3203	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
9		resfunexg 6479	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
10	7, 8, 9	mp2an 708	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝑥) ∈ V
11		rneq 5351	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → ran 𝑤 = ran (𝐹 ↾ 𝑥))
12		df-ima 5127	. . . . . . . . . . . . 13 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
13	11, 12	syl6eqr 2674	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → ran 𝑤 = (𝐹 “ 𝑥))
14	13	difeq2d 3728	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → (𝐴 ∖ ran 𝑤) = (𝐴 ∖ (𝐹 “ 𝑥)))
15	14	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → (𝐺‘(𝐴 ∖ ran 𝑤)) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
16		rneq 5351	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ran 𝑧 = ran 𝑤)
17	16	difeq2d 3728	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐴 ∖ ran 𝑧) = (𝐴 ∖ ran 𝑤))
18	17	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝐺‘(𝐴 ∖ ran 𝑧)) = (𝐺‘(𝐴 ∖ ran 𝑤)))
19	18	cbvmptv 4750	. . . . . . . . . 10 ⊢ (𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))) = (𝑤 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑤)))
20		fvex 6201	. . . . . . . . . 10 ⊢ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ V
21	15, 19, 20	fvmpt 6282	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝑥) ∈ V → ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(𝐹 ↾ 𝑥)) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
22	10, 21	ax-mp 5	. . . . . . . 8 ⊢ ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(𝐹 ↾ 𝑥)) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥)))
23	3	reseq1i 5392	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)
24	23	fveq2i 6194	. . . . . . . 8 ⊢ ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(𝐹 ↾ 𝑥)) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
25	22, 24	eqtr3i 2646	. . . . . . 7 ⊢ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
26	2, 4, 25	3eqtr4g 2681	. . . . . 6 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
27	26	ad2antlr 763	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐹‘𝑥) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
28		difss 3737	. . . . . . . . 9 ⊢ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴
29		elpw2g 4827	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴))
30	1, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴))
31	28, 30	mpbiri 248	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴)
32		dnnumch.g	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
33		neeq1 2856	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅))
34		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐺‘𝑦) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
35		id 22	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → 𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)))
36	34, 35	eleq12d 2695	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝐺‘𝑦) ∈ 𝑦 ↔ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
37	33, 36	imbi12d 334	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
38	37	rspcva 3307	. . . . . . . 8 ⊢ (((𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
39	31, 32, 38	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
40	39	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ On) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
41	40	imp 445	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
42	27, 41	eqeltrd 2701	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
43	42	ex 450	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ On) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
44	43	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
45	5	tz7.49c 7541	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
46	1, 44, 45	syl2anc 693	1 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   ↦ cmpt 4729  ran crn 5115   ↾ cres 5116   “ cima 5117  Oncon0 5723  Fun wfun 5882   Fn wfn 5883  –1-1-onto→wf1o 5887  ‘cfv 5888  recscrecs 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  dnnumch2  37615