Theorem ac5num 9958

Description: A version of ac5b 10400 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 7717	. . . 4 ⊢ (∪ 𝐴 ∈ dom card → 𝐴 ∈ V)
2		dfac8b 9953	. . . 4 ⊢ (∪ 𝐴 ∈ dom card → ∃𝑟 𝑟 We ∪ 𝐴)
3		dfac8c 9955	. . . 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 727	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → 𝐴 ∈ V)
7	6	mptexd 7179	. . 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 715	. . . . . . . . . 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 3149	. . . . . . . 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 6840	. . . . . . . . 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 3563	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥 ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ 𝑦)
19	14, 18	sylan 581	. . . . . 6 ⊢ ((((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ 𝑦)
20		elunii 4855	. . . . . 6 ⊢ (((𝑔‘𝑦) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ ∪ 𝐴)
21	19, 20	sylancom 589	. . . . 5 ⊢ ((((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ ∪ 𝐴)
22	21	fmpttd 7067	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴)
23		fveq2 6840	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑔‘𝑦) = (𝑔‘𝑥))
24		eqid 2736	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))
25		fvex 6853	. . . . . . . 8 ⊢ (𝑔‘𝑥) ∈ V
26	23, 24, 25	fvmpt 6947	. . . . . . 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 3081	. . . . 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 6646	. . . 4 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (𝑓:𝐴⟶∪ 𝐴 ↔ (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴))
32		fveq1 6839	. . . . . 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 3160	. . . 4 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥))
35	31, 34	anbi12d 633	. . 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 3540	. 2 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
37	5, 36	exlimddv 1937	1 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  Vcvv 3429  ∅c0 4273  ∪ cuni 4850   ↦ cmpt 5166   We wwe 5583  dom cdm 5631  ⟶wf 6494  ‘cfv 6498  cardccrd 9859

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-en 8894  df-card 9863

This theorem is referenced by:  numacn  9971  ac5b  10400  ac6num  10401