Theorem numacn 10087

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 3499	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		simpll 767	. . . . . . . 8 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑋 ∈ dom card)
3		elmapi 8888	. . . . . . . . . . . 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 6745	. . . . . . . . . 10 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}))
6	5	difss2d 4149	. . . . . . . . 9 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ 𝒫 𝑋)
7		sspwuni 5105	. . . . . . . . 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 10077	. . . . . . . 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 4492	. . . . . . . . 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 4716	. . . . . . . 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 10074	. . . . . . 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 776	. . . . . . 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 6738	. . . . . . . . . 10 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓 Fn 𝐴)
19		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑓‘𝑥) → (ℎ‘𝑦) = (ℎ‘(𝑓‘𝑥)))
20		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑓‘𝑥) → 𝑦 = (𝑓‘𝑥))
21	19, 20	eleq12d 2833	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑓‘𝑥) → ((ℎ‘𝑦) ∈ 𝑦 ↔ (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
22	21	ralrn 7108	. . . . . . . . . 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 10086	. . . . . . 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 1933	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
29	28	ralrimiva 3144	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
30		isacn 10082	. . . 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 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605  {csn 4631  ∪ cuni 4912  dom cdm 5689  ran crn 5690   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865  cardccrd 9973  AC wacn 9976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-card 9977  df-acn 9980

This theorem is referenced by:  acnnum  10090  fodomnum  10095  acacni  10179  dfac13  10181  iundom  10580  iunctb  10612