Theorem dfac8alem 10029

Description: Lemma for dfac8a 10030. 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 3478	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		difss 4090	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴
3		elpw2g 5306	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → ((𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴))
4	2, 3	mpbiri 261	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (𝐴 ∖ (𝐹 “ 𝑥)) ∈ 𝒫 𝐴)
5		neeq1 3022	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅))
6		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → (𝑔‘𝑦) = (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))))
7		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → 𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)))
8	6, 7	eleq12d 2859	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑔‘𝑦) ∈ 𝑦 ↔ (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
9	5, 8	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐴 ∖ (𝐹 “ 𝑥)) → ((𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
10	9	rspcv 3579	. . . . . . . . . . 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 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
12	11	3imp 1128	. . . . . . . . 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 8391	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))
15	13	tfr1 8390	. . . . . . . . . . . . . 14 ⊢ 𝐹 Fn On
16		fnfun 6639	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn On → Fun 𝐹)
17	15, 16	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 𝐹
18		vex 3461	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
19		resfunexg 7220	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
20	17, 18, 19	mp2an 705	. . . . . . . . . . . 12 ⊢ (𝐹 ↾ 𝑥) ∈ V
21		rneq 5928	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = ran (𝐹 ↾ 𝑥))
22		df-ima 5676	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
23	21, 22	eqtr4di 2818	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = (𝐹 “ 𝑥))
24	23	difeq2d 4081	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → (𝐴 ∖ ran 𝑓) = (𝐴 ∖ (𝐹 “ 𝑥)))
25	24	fveq2d 6889	. . . . . . . . . . . . 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 6898	. . . . . . . . . . . . 13 ⊢ (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ V
28	25, 26, 27	fvmpt 6993	. . . . . . . . . . . 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 2816	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))))
31	30	eleq1d 2850	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) ↔ (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
32	12, 31	syl5ibrcom 250	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ (𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅) → (𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
33	32	3expia 1139	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝑥 ∈ On → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
34	33	com23 87	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → (𝑥 ∈ On → ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
35	34	ralrimiv 3158	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦)) → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
36	35	ex 418	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))))
37	15	tz7.49c 8439	. . . . . 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 418	. . . . 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 8975	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝑥 ≈ 𝐴)
40		isnumi 9948	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑥 ≈ 𝐴) → 𝐴 ∈ dom card)
41	39, 40	sylan2 605	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴) → 𝐴 ∈ dom card)
42	41	rexlimiva 3160	. . . . 5 ⊢ (∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ∈ dom card)
43	38, 42	syl6 36	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → 𝐴 ∈ dom card))
44	36, 43	syld 48	. . 3 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
45	1, 44	syl 18	. 2 ⊢ (𝐴 ∈ 𝐶 → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
46	45	exlimdv 1966	1 ⊢ (𝐴 ∈ 𝐶 → (∃𝑔∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564   class class class wbr 5111   ↦ cmpt 5194  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666  Oncon0 6364  Fun wfun 6534   Fn wfn 6535  –1-1-onto→wf1o 6539  ‘cfv 6540  recscrecs 8363   ≈ cen 8946  cardccrd 9937

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-en 8950  df-card 9941

This theorem is used by:  dfac8a  10030