Theorem dnnumch1 43870

Description: Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 10036. (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 8396	. . . . . . 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 6883	. . . . . . 7 ⊢ (𝐹‘𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))‘𝑥)
5	3	tfr1 8389	. . . . . . . . . . 11 ⊢ 𝐹 Fn On
6		fnfun 6636	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → Fun 𝐹)
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 𝐹
8		vex 3457	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
9		resfunexg 7217	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
10	7, 8, 9	mp2an 705	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝑥) ∈ V
11		rneq 5924	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → ran 𝑤 = ran (𝐹 ↾ 𝑥))
12		df-ima 5672	. . . . . . . . . . . . 13 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
13	11, 12	eqtr4di 2815	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → ran 𝑤 = (𝐹 “ 𝑥))
14	13	difeq2d 4077	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → (𝐴 ∖ ran 𝑤) = (𝐴 ∖ (𝐹 “ 𝑥)))
15	14	fveq2d 6886	. . . . . . . . . 10 ⊢ (𝑤 = (𝐹 ↾ 𝑥) → (𝐺‘(𝐴 ∖ ran 𝑤)) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
16		rneq 5924	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ran 𝑧 = ran 𝑤)
17	16	difeq2d 4077	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐴 ∖ ran 𝑧) = (𝐴 ∖ ran 𝑤))
18	17	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝐺‘(𝐴 ∖ ran 𝑧)) = (𝐺‘(𝐴 ∖ ran 𝑤)))
19	18	cbvmptv 5213	. . . . . . . . . 10 ⊢ (𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))) = (𝑤 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑤)))
20		fvex 6895	. . . . . . . . . 10 ⊢ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ V
21	15, 19, 20	fvmpt 6990	. . . . . . . . 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 5972	. . . . . . . . 9 ⊢ (𝐹 ↾ 𝑥) = (recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥)
24	23	fveq2i 6885	. . . . . . . 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 2787	. . . . . . 7 ⊢ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) = ((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))‘(recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) ↾ 𝑥))
26	2, 4, 25	3eqtr4g 2822	. . . . . 6 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
27	26	ad2antlr 740	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐹‘𝑥) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
28		difss 4086	. . . . . . . . 9 ⊢ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴
29		elpw2g 5302	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴))
30	1, 29	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴))
31	28, 30	mpbiri 261	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴)
32		dnnumch.g	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
33		neeq1 3019	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅))
34		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐺‘𝑦) = (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))))
35		id 23	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → 𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)))
36	34, 35	eleq12d 2856	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝐺‘𝑦) ∈ 𝑦 ↔ (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
37	33, 36	imbi12d 347	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
38	37	rspcva 3577	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
40	39	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ On) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
41	40	imp 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐺‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
42	27, 41	eqeltrd 2862	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ On) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
43	42	ex 418	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ On) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
44	43	ralrimiva 3156	. 2 ⊢ (𝜑 → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
45	5	tz7.49c 8438	. 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 596	1 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560   ↦ cmpt 5190  ran crn 5660   ↾ cres 5661   “ cima 5662  Oncon0 6361  Fun wfun 6531   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘cfv 6537  recscrecs 8362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  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 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363

This theorem is used by:  dnnumch2  43871