Theorem dfac8alem 9920

Description: Lemma for dfac8a 9921. If the power set of a set has a choice function, then the set is numerable. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)

Hypotheses

Ref	Expression
dfac8alem.2	⊢ 𝐹 = recs(𝐺)
dfac8alem.3	⊢ 𝐺 = (𝑓 ∈ V ↦ (𝑔‘(𝐴 ∖ ran 𝑓)))

Assertion

Ref	Expression
dfac8alem	⊢ (𝐴 ∈ 𝐶 → (∃𝑔∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Distinct variable groups:   𝑓,𝑔,𝑦,𝐴   𝐶,𝑔   𝑓,𝐹,𝑦

Allowed substitution hints:   𝐶(𝑦,𝑓)   𝐹(𝑔)   𝐺(𝑦,𝑓,𝑔)

Proof of Theorem dfac8alem

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3457	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		difss 4083	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴
3		elpw2g 5269	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → ((𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴))
4	2, 3	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴)
5		neeq1 2990	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅))
6		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑔‘𝑦) = (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))))
7		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → 𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)))
8	6, 7	eleq12d 2825	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
9	5, 8	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
10	9	rspcv 3568	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴 → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
11	4, 10	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
12	11	3imp 1110	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))
13		dfac8alem.2	. . . . . . . . . . . 12 ⊢ 𝐹 = recs(𝐺)
14	13	tfr2 8317	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))
15	13	tfr1 8316	. . . . . . . . . . . . . 14 ⊢ 𝐹 Fn On
16		fnfun 6581	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn On → Fun 𝐹)
17	15, 16	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 𝐹
18		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
19		resfunexg 7149	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
20	17, 18, 19	mp2an 692	. . . . . . . . . . . 12 ⊢ (𝐹 ↾ 𝑥) ∈ V
21		rneq 5875	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = ran (𝐹 ↾ 𝑥))
22		df-ima 5627	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
23	21, 22	eqtr4di 2784	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = (𝐹 “ 𝑥))
24	23	difeq2d 4073	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝐴 ∖ ran 𝑓) = (𝐴 ∖ (𝐹 “ 𝑥)))
25	24	fveq2d 6826	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝑔‘(𝐴 ∖ ran 𝑓)) = (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))))
26		dfac8alem.3	. . . . . . . . . . . . 13 ⊢ 𝐺 = (𝑓 ∈ V ↦ (𝑔‘(𝐴 ∖ ran 𝑓)))
27		fvex 6835	. . . . . . . . . . . . 13 ⊢ (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ V
28	25, 26, 27	fvmpt 6929	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ 𝑥) ∈ V → (𝐺‘(𝐹 ↾ 𝑥)) = (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))))
29	20, 28	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐺‘(𝐹 ↾ 𝑥)) = (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥)))
30	14, 29	eqtrdi 2782	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))))
31	30	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) ↔ (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
32	12, 31	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
33	32	3expia 1121	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
34	33	com23 86	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → (𝑥 ∈ On → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
35	34	ralrimiv 3123	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
36	35	ex 412	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
37	15	tz7.49c 8365	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
38	37	ex 412	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴))
39	18	f1oen 8895	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝑥 ≈ 𝐴)
40		isnumi 9839	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card)
41	39, 40	sylan2 593	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴) → 𝐴 ∈ dom card)
42	41	rexlimiva 3125	. . . . 5 ⊢ (∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ∈ dom card)
43	38, 42	syl6 35	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → 𝐴 ∈ dom card))
44	36, 43	syld 47	. . 3 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
45	1, 44	syl 17	. 2 ⊢ (𝐴 ∈ 𝐶 → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
46	45	exlimdv 1934	1 ⊢ (𝐴 ∈ 𝐶 → (∃𝑔∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∖ cdif 3894   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547   class class class wbr 5089   ↦ cmpt 5170  dom cdm 5614  ran crn 5615   ↾ cres 5616   “ cima 5617  Oncon0 6306  Fun wfun 6475   Fn wfn 6476  –1-1-onto→wf1o 6480  ‘cfv 6481  recscrecs 8290   ≈ cen 8866  cardccrd 9828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-en 8870  df-card 9832

This theorem is referenced by:  dfac8a  9921