Theorem ac5num 10027

Description: A version of ac5b 10469 with the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)

Assertion

Ref	Expression
ac5num	⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))

Distinct variable group:   𝑥,𝑓,𝐴

Proof of Theorem ac5num

Dummy variables 𝑔 𝑟 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniexr 7746	. . . 4 ⊢ (∪ 𝐴 ∈ dom card → 𝐴 ∈ V)
2		dfac8b 10022	. . . 4 ⊢ (∪ 𝐴 ∈ dom card → ∃𝑟 𝑟 We ∪ 𝐴)
3		dfac8c 10024	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑟 𝑟 We ∪ 𝐴 → ∃𝑔∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)))
4	1, 2, 3	sylc 65	. . 3 ⊢ (∪ 𝐴 ∈ dom card → ∃𝑔∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥))
5	4	adantr 481	. 2 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥))
6	1	ad2antrr 724	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → 𝐴 ∈ V)
7	6	mptexd 7222	. . 3 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) ∈ V)
8		nelne2 3040	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ ∅ ∈ 𝐴) → 𝑥 ≠ ∅)
9	8	ancoms 459	. . . . . . . . . . 11 ⊢ ((¬ ∅ ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅)
10	9	adantll 712	. . . . . . . . . 10 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅)
11		pm2.27 42	. . . . . . . . . 10 ⊢ (𝑥 ≠ ∅ → ((𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) → (𝑔‘𝑥) ∈ 𝑥))
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) → (𝑔‘𝑥) ∈ 𝑥))
13	12	ralimdva 3167	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥))
14	13	imp 407	. . . . . . 7 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥)
15		fveq2 6888	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
16		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
17	15, 16	eleq12d 2827	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑔‘𝑥) ∈ 𝑥 ↔ (𝑔‘𝑦) ∈ 𝑦))
18	17	rspccva 3611	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥 ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ 𝑦)
19	14, 18	sylan 580	. . . . . 6 ⊢ ((((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ 𝑦)
20		elunii 4912	. . . . . 6 ⊢ (((𝑔‘𝑦) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ ∪ 𝐴)
21	19, 20	sylancom 588	. . . . 5 ⊢ ((((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ ∪ 𝐴)
22	21	fmpttd 7111	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴)
23		fveq2 6888	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑔‘𝑦) = (𝑔‘𝑥))
24		eqid 2732	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))
25		fvex 6901	. . . . . . . 8 ⊢ (𝑔‘𝑥) ∈ V
26	23, 24, 25	fvmpt 6995	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) = (𝑔‘𝑥))
27	26	eleq1d 2818	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥 ↔ (𝑔‘𝑥) ∈ 𝑥))
28	27	ralbiia 3091	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥)
29	14, 28	sylibr 233	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥)
30	22, 29	jca 512	. . 3 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥))
31		feq1 6695	. . . 4 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (𝑓:𝐴⟶∪ 𝐴 ↔ (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴))
32		fveq1 6887	. . . . . 6 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (𝑓‘𝑥) = ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥))
33	32	eleq1d 2818	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → ((𝑓‘𝑥) ∈ 𝑥 ↔ ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥))
34	33	ralbidv 3177	. . . 4 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥))
35	31, 34	anbi12d 631	. . 3 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → ((𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) ↔ ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥)))
36	7, 30, 35	spcedv 3588	. 2 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
37	5, 36	exlimddv 1938	1 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  Vcvv 3474  ∅c0 4321  ∪ cuni 4907   ↦ cmpt 5230   We wwe 5629  dom cdm 5675  ⟶wf 6536  ‘cfv 6540  cardccrd 9926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-en 8936  df-card 9930

This theorem is referenced by:  numacn  10040  ac5b  10469  ac6num  10470