Theorem ac5num 10055

Description: A version of ac5b 10497 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 7762	. . . 4 ⊢ (∪ 𝐴 ∈ dom card → 𝐴 ∈ V)
2		dfac8b 10050	. . . 4 ⊢ (∪ 𝐴 ∈ dom card → ∃𝑟 𝑟 We ∪ 𝐴)
3		dfac8c 10052	. . . 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 7221	. . 3 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) ∈ V)
8		nelne2 3031	. . . . . . . . . . . 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 3153	. . . . . . . 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 6881	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
16		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
17	15, 16	eleq12d 2829	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑔‘𝑥) ∈ 𝑥 ↔ (𝑔‘𝑦) ∈ 𝑦))
18	17	rspccva 3605	. . . . . . 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 4893	. . . . . 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 7110	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴)
23		fveq2 6881	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑔‘𝑦) = (𝑔‘𝑥))
24		eqid 2736	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))
25		fvex 6894	. . . . . . . 8 ⊢ (𝑔‘𝑥) ∈ V
26	23, 24, 25	fvmpt 6991	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) = (𝑔‘𝑥))
27	26	eleq1d 2820	. . . . . 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 6691	. . . 4 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (𝑓:𝐴⟶∪ 𝐴 ↔ (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴))
32		fveq1 6880	. . . . . 6 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (𝑓‘𝑥) = ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥))
33	32	eleq1d 2820	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → ((𝑓‘𝑥) ∈ 𝑥 ↔ ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥))
34	33	ralbidv 3164	. . . 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 3582	. 2 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
37	5, 36	exlimddv 1935	1 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  Vcvv 3464  ∅c0 4313  ∪ cuni 4888   ↦ cmpt 5206   We wwe 5610  dom cdm 5659  ⟶wf 6532  ‘cfv 6536  cardccrd 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-en 8965  df-card 9958

This theorem is referenced by:  numacn  10068  ac5b  10497  ac6num  10498