Theorem numacn 9969

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 3453	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		simpll 772	. . . . . . . 8 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑋 ∈ dom card)
3		elmapi 8793	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
4	3	adantl 482	. . . . . . . . . . 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 4076	. . . . . . . . 9 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ 𝒫 𝑋)
7		sspwuni 5036	. . . . . . . . 9 ⊢ (ran 𝑓 ⊆ 𝒫 𝑋 ↔ ∪ ran 𝑓 ⊆ 𝑋)
8	6, 7	sylib 219	. . . . . . . 8 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∪ ran 𝑓 ⊆ 𝑋)
9		ssnum 9959	. . . . . . . 8 ⊢ ((𝑋 ∈ dom card ∧ ∪ ran 𝑓 ⊆ 𝑋) → ∪ ran 𝑓 ∈ dom card)
10	2, 8, 9	syl2anc 590	. . . . . . 7 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∪ ran 𝑓 ∈ dom card)
11		ssdifin0 4420	. . . . . . . . 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 4650	. . . . . . . 8 ⊢ ((ran 𝑓 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran 𝑓)
14	12, 13	sylib 219	. . . . . . 7 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ¬ ∅ ∈ ran 𝑓)
15		ac5num 9956	. . . . . . 7 ⊢ ((∪ ran 𝑓 ∈ dom card ∧ ¬ ∅ ∈ ran 𝑓) → ∃ℎ(ℎ:ran 𝑓⟶∪ ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦))
16	10, 14, 15	syl2anc 590	. . . . . 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 781	. . . . . . 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 2834	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑓‘𝑥) → ((ℎ‘𝑦) ∈ 𝑦 ↔ (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
22	21	ralrn 7036	. . . . . . . . . 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 477	. . . . . . . 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 722	. . . . . . 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 9968	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
27	17, 25, 26	syl2anc 590	. . . . . 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 1942	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
29	28	ralrimiva 3132	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
30		isacn 9964	. . . 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 258	. . 3 ⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → 𝑋 ∈ AC 𝐴)
32	31	expcom 414	. 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 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536  {csn 4562  ∪ cuni 4845  dom cdm 5625  ran crn 5626   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ↑_m cmap 8770  cardccrd 9857  AC wacn 9860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-card 9861  df-acn 9864

This theorem is referenced by:  acnnum  9972  fodomnum  9977  acacni  10061  dfac13  10063  iundom  10462  iunctb  10495