Theorem dnnumch1 43035

Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 10049. (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 8423	. . . . . . 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 6882	. . . . . . 7 ⊢ (𝐹‘𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥)
5	3	tfr1 8416	. . . . . . . . . . 11 ⊢ 𝐹 Fn On
6		fnfun 6643	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → Fun 𝐹)
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 𝐹
8		vex 3468	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
9		resfunexg 7212	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
10	7, 8, 9	mp2an 692	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝑥) ∈ V
11		rneq 5921	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → ran 𝑤 = ran (𝐹 ↾ 𝑥))
12		df-ima 5672	. . . . . . . . . . . . 13 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
13	11, 12	eqtr4di 2789	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → ran 𝑤 = (𝐹 “ 𝑥))
14	13	difeq2d 4106	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → (𝐴 ∖ ran 𝑤) = (𝐴 ∖ (𝐹 “ 𝑥)))
15	14	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → (𝐺‘(𝐴 ∖ ran 𝑤)) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
16		rneq 5921	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ran 𝑧 = ran 𝑤)
17	16	difeq2d 4106	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐴 ∖ ran 𝑧) = (𝐴 ∖ ran 𝑤))
18	17	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝐺‘(𝐴 ∖ ran 𝑧)) = (𝐺‘(𝐴 ∖ ran 𝑤)))
19	18	cbvmptv 5230	. . . . . . . . . 10 ⊢ (𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))) = (𝑤 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑤)))
20		fvex 6894	. . . . . . . . . 10 ⊢ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ V
21	15, 19, 20	fvmpt 6991	. . . . . . . . 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 5967	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)
24	23	fveq2i 6884	. . . . . . . 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 2761	. . . . . . 7 ⊢ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
26	2, 4, 25	3eqtr4g 2796	. . . . . 6 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
27	26	ad2antlr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐹‘𝑥) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
28		difss 4116	. . . . . . . . 9 ⊢ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴
29		elpw2g 5308	. . . . . . . . . 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 258	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴)
32		dnnumch.g	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
33		neeq1 2995	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅))
34		fveq2 6881	. . . . . . . . . . 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 2829	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝐺‘𝑦) ∈ 𝑦 ↔ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
37	33, 36	imbi12d 344	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
38	37	rspcva 3604	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
40	39	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ On) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
41	40	imp 406	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
42	27, 41	eqeltrd 2835	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
43	42	ex 412	. . 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 8465	. 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 584	1 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∖ cdif 3928   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580   ↦ cmpt 5206  ran crn 5660   ↾ cres 5661   “ cima 5662  Oncon0 6357  Fun wfun 6530   Fn wfn 6531  –1-1-onto→wf1o 6535  ‘cfv 6536  recscrecs 8389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390

This theorem is referenced by:  dnnumch2  43036