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Theorem numacn 10089
Description: A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
numacn (𝐴𝑉 → (𝑋 ∈ dom card → 𝑋AC 𝐴))

Proof of Theorem numacn
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3501 . 2 (𝐴𝑉𝐴 ∈ V)
2 simpll 767 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑋 ∈ dom card)
3 elmapi 8889 . . . . . . . . . . . 12 (𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
43adantl 481 . . . . . . . . . . 11 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
54frnd 6744 . . . . . . . . . 10 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}))
65difss2d 4139 . . . . . . . . 9 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ 𝒫 𝑋)
7 sspwuni 5100 . . . . . . . . 9 (ran 𝑓 ⊆ 𝒫 𝑋 ran 𝑓𝑋)
86, 7sylib 218 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓𝑋)
9 ssnum 10079 . . . . . . . 8 ((𝑋 ∈ dom card ∧ ran 𝑓𝑋) → ran 𝑓 ∈ dom card)
102, 8, 9syl2anc 584 . . . . . . 7 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ∈ dom card)
11 ssdifin0 4486 . . . . . . . . 9 (ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}) → (ran 𝑓 ∩ {∅}) = ∅)
125, 11syl 17 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → (ran 𝑓 ∩ {∅}) = ∅)
13 disjsn 4711 . . . . . . . 8 ((ran 𝑓 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran 𝑓)
1412, 13sylib 218 . . . . . . 7 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ¬ ∅ ∈ ran 𝑓)
15 ac5num 10076 . . . . . . 7 (( ran 𝑓 ∈ dom card ∧ ¬ ∅ ∈ ran 𝑓) → ∃(:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦))
1610, 14, 15syl2anc 584 . . . . . 6 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃(:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦))
17 simpllr 776 . . . . . . 7 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦)) → 𝐴 ∈ V)
184ffnd 6737 . . . . . . . . . 10 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓 Fn 𝐴)
19 fveq2 6906 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑥) → (𝑦) = (‘(𝑓𝑥)))
20 id 22 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑥) → 𝑦 = (𝑓𝑥))
2119, 20eleq12d 2835 . . . . . . . . . . 11 (𝑦 = (𝑓𝑥) → ((𝑦) ∈ 𝑦 ↔ (‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2221ralrn 7108 . . . . . . . . . 10 (𝑓 Fn 𝐴 → (∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦 ↔ ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2318, 22syl 17 . . . . . . . . 9 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → (∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦 ↔ ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2423biimpa 476 . . . . . . . 8 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦) → ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥))
2524adantrl 716 . . . . . . 7 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦)) → ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥))
26 acnlem 10088 . . . . . . 7 ((𝐴 ∈ V ∧ ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
2717, 25, 26syl2anc 584 . . . . . 6 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
2816, 27exlimddv 1935 . . . . 5 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
2928ralrimiva 3146 . . . 4 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
30 isacn 10084 . . . 4 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → (𝑋AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
3129, 30mpbird 257 . . 3 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → 𝑋AC 𝐴)
3231expcom 413 . 2 (𝐴 ∈ V → (𝑋 ∈ dom card → 𝑋AC 𝐴))
331, 32syl 17 1 (𝐴𝑉 → (𝑋 ∈ dom card → 𝑋AC 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  Vcvv 3480  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  cardccrd 9975  AC wacn 9978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-card 9979  df-acn 9982
This theorem is referenced by:  acnnum  10092  fodomnum  10097  acacni  10181  dfac13  10183  iundom  10582  iunctb  10614
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