Theorem dnnumch1 43504

Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 9947. (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 8337	. . . . . . 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 6832	. . . . . . 7 ⊢ (𝐹‘𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥)
5	3	tfr1 8330	. . . . . . . . . . 11 ⊢ 𝐹 Fn On
6		fnfun 6589	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → Fun 𝐹)
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 𝐹
8		vex 3437	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
9		resfunexg 7163	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
10	7, 8, 9	mp2an 699	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝑥) ∈ V
11		rneq 5885	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → ran 𝑤 = ran (𝐹 ↾ 𝑥))
12		df-ima 5634	. . . . . . . . . . . . 13 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
13	11, 12	eqtr4di 2794	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → ran 𝑤 = (𝐹 “ 𝑥))
14	13	difeq2d 4060	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → (𝐴 ∖ ran 𝑤) = (𝐴 ∖ (𝐹 “ 𝑥)))
15	14	fveq2d 6835	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → (𝐺‘(𝐴 ∖ ran 𝑤)) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
16		rneq 5885	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ran 𝑧 = ran 𝑤)
17	16	difeq2d 4060	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐴 ∖ ran 𝑧) = (𝐴 ∖ ran 𝑤))
18	17	fveq2d 6835	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝐺‘(𝐴 ∖ ran 𝑧)) = (𝐺‘(𝐴 ∖ ran 𝑤)))
19	18	cbvmptv 5179	. . . . . . . . . 10 ⊢ (𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))) = (𝑤 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑤)))
20		fvex 6844	. . . . . . . . . 10 ⊢ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ V
21	15, 19, 20	fvmpt 6939	. . . . . . . . 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 5934	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)
24	23	fveq2i 6834	. . . . . . . 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 2766	. . . . . . 7 ⊢ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
26	2, 4, 25	3eqtr4g 2801	. . . . . 6 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
27	26	ad2antlr 734	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐹‘𝑥) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
28		difss 4069	. . . . . . . . 9 ⊢ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴
29		elpw2g 5264	. . . . . . . . . 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 260	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴)
32		dnnumch.g	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
33		neeq1 2998	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅))
34		fveq2 6831	. . . . . . . . . . 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 2835	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝐺‘𝑦) ∈ 𝑦 ↔ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
37	33, 36	imbi12d 346	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
38	37	rspcva 3560	. . . . . . . 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 591	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
40	39	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ On) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
41	40	imp 408	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
42	27, 41	eqeltrd 2841	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
43	42	ex 414	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ On) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
44	43	ralrimiva 3133	. 2 ⊢ (𝜑 → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
45	5	tz7.49c 8379	. 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 591	1 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  ∃wrex 3065  Vcvv 3433   ∖ cdif 3882   ⊆ wss 3885  ∅c0 4264  𝒫 cpw 4532   ↦ cmpt 5156  ran crn 5622   ↾ cres 5623   “ cima 5624  Oncon0 6314  Fun wfun 6483   Fn wfn 6484  –1-1-onto→wf1o 6488  ‘cfv 6489  recscrecs 8304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305

This theorem is referenced by:  dnnumch2  43505