Theorem dfac8alem 10108

Description: Lemma for dfac8a 10109. 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 3472	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		difss 4083	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ (𝐹 “ 𝑥)) ⊆ 𝐴
3		elpw2g 5295	. . . . . . . . . . . 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 3018	. . . . . . . . . . . . 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 2855	. . . . . . . . . . . . 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 3573	. . . . . . . . . . 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 8406	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ 𝑥)))
15	13	tfr1 8405	. . . . . . . . . . . . . 14 ⊢ 𝐹 Fn On
16		fnfun 6639	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn On → Fun 𝐹)
17	15, 16	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 𝐹
18		vex 3455	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
19		resfunexg 7221	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ V) → (𝐹 ↾ 𝑥) ∈ V)
20	17, 18, 19	mp2an 705	. . . . . . . . . . . 12 ⊢ (𝐹 ↾ 𝑥) ∈ V
21		rneq 5918	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = ran (𝐹 ↾ 𝑥))
22		df-ima 5664	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
23	21, 22	eqtr4di 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝐹 ↾ 𝑥) → ran 𝑓 = (𝐹 “ 𝑥))
24	23	difeq2d 4074	. . . . . . . . . . . . . 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 2812	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (𝐹‘𝑥) = (𝑔‘(𝐴 ∖ (𝐹 “ 𝑥))))
31	30	eleq1d 2846	. . . . . . . . 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 3154	. . . . 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 8456	. . . . . 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 8999	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝑥 ≈ 𝐴)
40		isnumi 10027	. . . . . . 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 3156	. . . . 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 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654  Oncon0 6362  Fun wfun 6532   Fn wfn 6533  –1-1-onto→wf1o 6537  ‘cfv 6538  recscrecs 8378   ≈ cen 8970  cardccrd 10016

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-en 8974  df-card 10020

This theorem is used by:  dfac8a  10109