Theorem numacn 9959

Description: A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
numacn	⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ dom card → 𝑋 ∈ AC 𝐴))

Proof of Theorem numacn

Dummy variables 𝑓 𝑔 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3461	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		simpll 766	. . . . . . . 8 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑋 ∈ dom card)
3		elmapi 8786	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
4	3	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
5	4	frnd 6670	. . . . . . . . . 10 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}))
6	5	difss2d 4091	. . . . . . . . 9 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ 𝒫 𝑋)
7		sspwuni 5055	. . . . . . . . 9 ⊢ (ran 𝑓 ⊆ 𝒫 𝑋 ↔ ∪ ran 𝑓 ⊆ 𝑋)
8	6, 7	sylib 218	. . . . . . . 8 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∪ ran 𝑓 ⊆ 𝑋)
9		ssnum 9949	. . . . . . . 8 ⊢ ((𝑋 ∈ dom card ∧ ∪ ran 𝑓 ⊆ 𝑋) → ∪ ran 𝑓 ∈ dom card)
10	2, 8, 9	syl2anc 584	. . . . . . 7 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∪ ran 𝑓 ∈ dom card)
11		ssdifin0 4438	. . . . . . . . 9 ⊢ (ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}) → (ran 𝑓 ∩ {∅}) = ∅)
12	5, 11	syl 17	. . . . . . . 8 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → (ran 𝑓 ∩ {∅}) = ∅)
13		disjsn 4668	. . . . . . . 8 ⊢ ((ran 𝑓 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran 𝑓)
14	12, 13	sylib 218	. . . . . . 7 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ¬ ∅ ∈ ran 𝑓)
15		ac5num 9946	. . . . . . 7 ⊢ ((∪ ran 𝑓 ∈ dom card ∧ ¬ ∅ ∈ ran 𝑓) → ∃ℎ(ℎ:ran 𝑓⟶∪ ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦))
16	10, 14, 15	syl2anc 584	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃ℎ(ℎ:ran 𝑓⟶∪ ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦))
17		simpllr 775	. . . . . . 7 ⊢ ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (ℎ:ran 𝑓⟶∪ ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦)) → 𝐴 ∈ V)
18	4	ffnd 6663	. . . . . . . . . 10 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓 Fn 𝐴)
19		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑓‘𝑥) → (ℎ‘𝑦) = (ℎ‘(𝑓‘𝑥)))
20		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑓‘𝑥) → 𝑦 = (𝑓‘𝑥))
21	19, 20	eleq12d 2830	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑓‘𝑥) → ((ℎ‘𝑦) ∈ 𝑦 ↔ (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
22	21	ralrn 7033	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝐴 → (∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
23	18, 22	syl 17	. . . . . . . . 9 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → (∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
24	23	biimpa 476	. . . . . . . 8 ⊢ ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦) → ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))
25	24	adantrl 716	. . . . . . 7 ⊢ ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (ℎ:ran 𝑓⟶∪ ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦)) → ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))
26		acnlem 9958	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
27	17, 25, 26	syl2anc 584	. . . . . 6 ⊢ ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (ℎ:ran 𝑓⟶∪ ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
28	16, 27	exlimddv 1936	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
29	28	ralrimiva 3128	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
30		isacn 9954	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
31	29, 30	mpbird 257	. . 3 ⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → 𝑋 ∈ AC 𝐴)
32	31	expcom 413	. 2 ⊢ (𝐴 ∈ V → (𝑋 ∈ dom card → 𝑋 ∈ AC 𝐴))
33	1, 32	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ dom card → 𝑋 ∈ AC 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  ∪ cuni 4863  dom cdm 5624  ran crn 5625   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  cardccrd 9847  AC wacn 9850

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-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  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-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-card 9851  df-acn 9854

This theorem is referenced by:  acnnum  9962  fodomnum  9967  acacni  10051  dfac13  10053  iundom  10452  iunctb  10485