Theorem ac5num 9946

Description: A version of ac5b 10388 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 7708	. . . 4 ⊢ (∪ 𝐴 ∈ dom card → 𝐴 ∈ V)
2		dfac8b 9941	. . . 4 ⊢ (∪ 𝐴 ∈ dom card → ∃𝑟 𝑟 We ∪ 𝐴)
3		dfac8c 9943	. . . 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 480	. 2 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥))
6	1	ad2antrr 726	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → 𝐴 ∈ V)
7	6	mptexd 7170	. . 3 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) ∈ V)
8		nelne2 3030	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ ∅ ∈ 𝐴) → 𝑥 ≠ ∅)
9	8	ancoms 458	. . . . . . . . . . 11 ⊢ ((¬ ∅ ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅)
10	9	adantll 714	. . . . . . . . . 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 3148	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥))
14	13	imp 406	. . . . . . 7 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥)
15		fveq2 6834	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
16		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
17	15, 16	eleq12d 2830	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑔‘𝑥) ∈ 𝑥 ↔ (𝑔‘𝑦) ∈ 𝑦))
18	17	rspccva 3575	. . . . . . 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 4868	. . . . . 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 7060	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴)
23		fveq2 6834	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑔‘𝑦) = (𝑔‘𝑥))
24		eqid 2736	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))
25		fvex 6847	. . . . . . . 8 ⊢ (𝑔‘𝑥) ∈ V
26	23, 24, 25	fvmpt 6941	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) = (𝑔‘𝑥))
27	26	eleq1d 2821	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥 ↔ (𝑔‘𝑥) ∈ 𝑥))
28	27	ralbiia 3080	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥)
29	14, 28	sylibr 234	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥)
30	22, 29	jca 511	. . 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 6640	. . . 4 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (𝑓:𝐴⟶∪ 𝐴 ↔ (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴))
32		fveq1 6833	. . . . . 6 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (𝑓‘𝑥) = ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥))
33	32	eleq1d 2821	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → ((𝑓‘𝑥) ∈ 𝑥 ↔ ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥))
34	33	ralbidv 3159	. . . 4 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥))
35	31, 34	anbi12d 632	. . 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 3552	. 2 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
37	5, 36	exlimddv 1936	1 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  Vcvv 3440  ∅c0 4285  ∪ cuni 4863   ↦ cmpt 5179   We wwe 5576  dom cdm 5624  ⟶wf 6488  ‘cfv 6492  cardccrd 9847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-en 8884  df-card 9851

This theorem is referenced by:  numacn  9959  ac5b  10388  ac6num  10389