Theorem dfac8alem 9166

Description: Lemma for dfac8a 9167. 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 3430	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		difss 3965	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴
3		elpw2g 5050	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → ((𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴))
4	2, 3	mpbiri 250	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴)
5		neeq1 3062	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅))
6		fveq2 6434	. . . . . . . . . . . . . 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 2901	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
9	5, 8	imbi12d 336	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
10	9	rspcv 3523	. . . . . . . . . . 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 1143	. . . . . . . . 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 7761	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))
15	13	tfr1 7760	. . . . . . . . . . . . . 14 ⊢ 𝐹 Fn On
16		fnfun 6222	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn On → Fun 𝐹)
17	15, 16	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 𝐹
18		vex 3418	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
19		resfunexg 6736	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
20	17, 18, 19	mp2an 685	. . . . . . . . . . . 12 ⊢ (𝐹 ↾ 𝑥) ∈ V
21		rneq 5584	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = ran (𝐹 ↾ 𝑥))
22		df-ima 5356	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
23	21, 22	syl6eqr 2880	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = (𝐹 “ 𝑥))
24	23	difeq2d 3956	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝐴 ∖ ran 𝑓) = (𝐴 ∖ (𝐹 “ 𝑥)))
25	24	fveq2d 6438	. . . . . . . . . . . . 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 6447	. . . . . . . . . . . . 13 ⊢ (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ V
28	25, 26, 27	fvmpt 6530	. . . . . . . . . . . 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	syl6eq 2878	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))))
31	30	eleq1d 2892	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) ↔ (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
32	12, 31	syl5ibrcom 239	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
33	32	3expia 1156	. . . . . . 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 3175	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
36	35	ex 403	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
37	15	tz7.49c 7808	. . . . . 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 403	. . . . 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 8244	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝑥 ≈ 𝐴)
40		isnumi 9086	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card)
41	39, 40	sylan2 588	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴) → 𝐴 ∈ dom card)
42	41	rexlimiva 3238	. . . . 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 2034	1 ⊢ (𝐴 ∈ 𝐶 → (∃𝑔∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1113   = wceq 1658  ∃wex 1880   ∈ wcel 2166   ≠ wne 3000  ∀wral 3118  ∃wrex 3119  Vcvv 3415   ∖ cdif 3796   ⊆ wss 3799  ∅c0 4145  𝒫 cpw 4379   class class class wbr 4874   ↦ cmpt 4953  dom cdm 5343  ran crn 5344   ↾ cres 5345   “ cima 5346  Oncon0 5964  Fun wfun 6118   Fn wfn 6119  –1-1-onto→wf1o 6123  ‘cfv 6124  recscrecs 7734   ≈ cen 8220  cardccrd 9075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-wrecs 7673  df-recs 7735  df-en 8224  df-card 9079

This theorem is referenced by:  dfac8a  9167