Theorem ac5num 8859

Description: A version of ac5b 9300 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 6972	. . . 4 ⊢ (∪ 𝐴 ∈ dom card → 𝐴 ∈ V)
2		dfac8b 8854	. . . 4 ⊢ (∪ 𝐴 ∈ dom card → ∃𝑟 𝑟 We ∪ 𝐴)
3		dfac8c 8856	. . . 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		nelne2 2891	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ ∅ ∈ 𝐴) → 𝑥 ≠ ∅)
7	6	ancoms 469	. . . . . . . . . . 11 ⊢ ((¬ ∅ ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅)
8	7	adantll 750	. . . . . . . . . 10 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅)
9		pm2.27 42	. . . . . . . . . 10 ⊢ (𝑥 ≠ ∅ → ((𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) → (𝑔‘𝑥) ∈ 𝑥))
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) → (𝑔‘𝑥) ∈ 𝑥))
11	10	ralimdva 2962	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥))
12	11	imp 445	. . . . . . 7 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥)
13		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑔‘𝑥) = (𝑔‘𝑦))
14		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
15	13, 14	eleq12d 2695	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑔‘𝑥) ∈ 𝑥 ↔ (𝑔‘𝑦) ∈ 𝑦))
16	15	rspccva 3308	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥 ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ 𝑦)
17	12, 16	sylan 488	. . . . . 6 ⊢ ((((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ 𝑦)
18		elunii 4441	. . . . . 6 ⊢ (((𝑔‘𝑦) ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ ∪ 𝐴)
19	17, 18	sylancom 701	. . . . 5 ⊢ ((((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) ∈ ∪ 𝐴)
20		eqid 2622	. . . . 5 ⊢ (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))
21	19, 20	fmptd 6385	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴)
22	1	ad2antrr 762	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → 𝐴 ∈ V)
23		elex 3212	. . . . 5 ⊢ (∪ 𝐴 ∈ dom card → ∪ 𝐴 ∈ V)
24	23	ad2antrr 762	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∪ 𝐴 ∈ V)
25		fex2 7121	. . . 4 ⊢ (((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴 ∧ 𝐴 ∈ V ∧ ∪ 𝐴 ∈ V) → (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) ∈ V)
26	21, 22, 24, 25	syl3anc 1326	. . 3 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) ∈ V)
27		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑔‘𝑦) = (𝑔‘𝑥))
28		fvex 6201	. . . . . . . 8 ⊢ (𝑔‘𝑥) ∈ V
29	27, 20, 28	fvmpt 6282	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) = (𝑔‘𝑥))
30	29	eleq1d 2686	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥 ↔ (𝑔‘𝑥) ∈ 𝑥))
31	30	ralbiia 2979	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝑥)
32	12, 31	sylibr 224	. . . 4 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥)
33	21, 32	jca 554	. . 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))
34		feq1 6026	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (𝑓:𝐴⟶∪ 𝐴 ↔ (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴))
35		fveq1 6190	. . . . . . 7 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (𝑓‘𝑥) = ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥))
36	35	eleq1d 2686	. . . . . 6 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → ((𝑓‘𝑥) ∈ 𝑥 ↔ ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥))
37	36	ralbidv 2986	. . . . 5 ⊢ (𝑓 = (𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥))
38	34, 37	anbi12d 747	. . . 4 ⊢ (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)))
39	38	spcegv 3294	. . 3 ⊢ ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)) ∈ V → (((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦)):𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝑦 ∈ 𝐴 ↦ (𝑔‘𝑦))‘𝑥) ∈ 𝑥) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)))
40	26, 33, 39	sylc 65	. 2 ⊢ (((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
41	5, 40	exlimddv 1863	1 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200  ∅c0 3915  ∪ cuni 4436   ↦ cmpt 4729   We wwe 5072  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-en 7956  df-card 8765

This theorem is referenced by:  numacn  8872  ac5b  9300  ac6num  9301