Theorem dfac8alem 10067

Description: Lemma for dfac8a 10068. 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 3499	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		difss 4146	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴
3		elpw2g 5339	. . . . . . . . . . . 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 3001	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅))
6		fveq2 6907	. . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . 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 3618	. . . . . . . . . . 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 8437	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))
15	13	tfr1 8436	. . . . . . . . . . . . . 14 ⊢ 𝐹 Fn On
16		fnfun 6669	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn On → Fun 𝐹)
17	15, 16	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 𝐹
18		vex 3482	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
19		resfunexg 7235	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
20	17, 18, 19	mp2an 692	. . . . . . . . . . . 12 ⊢ (𝐹 ↾ 𝑥) ∈ V
21		rneq 5950	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = ran (𝐹 ↾ 𝑥))
22		df-ima 5702	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
23	21, 22	eqtr4di 2793	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = (𝐹 “ 𝑥))
24	23	difeq2d 4136	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝐴 ∖ ran 𝑓) = (𝐴 ∖ (𝐹 “ 𝑥)))
25	24	fveq2d 6911	. . . . . . . . . . . . 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 6920	. . . . . . . . . . . . 13 ⊢ (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ V
28	25, 26, 27	fvmpt 7016	. . . . . . . . . . . 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 2791	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))))
31	30	eleq1d 2824	. . . . . . . . 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 1120	. . . . . . 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 3143	. . . . 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 8485	. . . . . 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 9012	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝑥 ≈ 𝐴)
40		isnumi 9984	. . . . . . 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 3145	. . . . 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 1931	1 ⊢ (𝐴 ∈ 𝐶 → (∃𝑔∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ∖ cdif 3960   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5689  ran crn 5690   ↾ cres 5691   “ cima 5692  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  –1-1-onto→wf1o 6562  ‘cfv 6563  recscrecs 8409   ≈ cen 8981  cardccrd 9973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-en 8985  df-card 9977

This theorem is referenced by:  dfac8a  10068