Theorem numacn 10109

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 3472	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		simpll 779	. . . . . . . 8 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑋 ∈ dom card)
3		elmapi 8853	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
4	3	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
5	4	frnd 6710	. . . . . . . . . 10 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}))
6	5	difss2d 4086	. . . . . . . . 9 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ 𝒫 𝑋)
7		sspwuni 5060	. . . . . . . . 9 ⊢ (ran 𝑓 ⊆ 𝒫 𝑋 ↔ ∪ ran 𝑓 ⊆ 𝑋)
8	6, 7	sylib 221	. . . . . . . 8 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∪ ran 𝑓 ⊆ 𝑋)
9		ssnum 10099	. . . . . . . 8 ⊢ ((𝑋 ∈ dom card ∧ ∪ ran 𝑓 ⊆ 𝑋) → ∪ ran 𝑓 ∈ dom card)
10	2, 8, 9	syl2anc 596	. . . . . . 7 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∪ ran 𝑓 ∈ dom card)
11		ssdifin0 4441	. . . . . . . . 9 ⊢ (ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}) → (ran 𝑓 ∩ {∅}) = ∅)
12	5, 11	syl 18	. . . . . . . 8 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → (ran 𝑓 ∩ {∅}) = ∅)
13		disjsn 4672	. . . . . . . 8 ⊢ ((ran 𝑓 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran 𝑓)
14	12, 13	sylib 221	. . . . . . 7 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ¬ ∅ ∈ ran 𝑓)
15		ac5num 10096	. . . . . . 7 ⊢ ((∪ ran 𝑓 ∈ dom card ∧ ¬ ∅ ∈ ran 𝑓) → ∃ℎ(ℎ:ran 𝑓⟶∪ ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦))
16	10, 14, 15	syl2anc 596	. . . . . 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 788	. . . . . . 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 6702	. . . . . . . . . 10 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓 Fn 𝐴)
19		fveq2 6877	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑓‘𝑥) → (ℎ‘𝑦) = (ℎ‘(𝑓‘𝑥)))
20		id 23	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑓‘𝑥) → 𝑦 = (𝑓‘𝑥))
21	19, 20	eleq12d 2855	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑓‘𝑥) → ((ℎ‘𝑦) ∈ 𝑦 ↔ (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
22	21	ralrn 7080	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝐴 → (∀𝑦 ∈ ran 𝑓(ℎ‘𝑦) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
23	18, 22	syl 18	. . . . . . . . 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 482	. . . . . . . 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 729	. . . . . . 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 10108	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 (ℎ‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
27	17, 25, 26	syl2anc 596	. . . . . 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 1968	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
29	28	ralrimiva 3155	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
30		isacn 10104	. . . 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 260	. . 3 ⊢ ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → 𝑋 ∈ AC 𝐴)
32	31	expcom 419	. 2 ⊢ (𝐴 ∈ V → (𝑋 ∈ dom card → 𝑋 ∈ AC 𝐴))
33	1, 32	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ dom card → 𝑋 ∈ AC 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867  dom cdm 5651  ran crn 5652   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831  cardccrd 9997  AC wacn 10000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-card 10001  df-acn 10004

This theorem is used by:  acnnum  10112  fodomnum  10117  acacni  10200  dfac13  10202  iundom  10607  iunctb  10640