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Theorem numacn 9463
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 3510 . 2 (𝐴𝑉𝐴 ∈ V)
2 simpll 763 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑋 ∈ dom card)
3 elmapi 8417 . . . . . . . . . . . 12 (𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
43adantl 482 . . . . . . . . . . 11 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
54frnd 6514 . . . . . . . . . 10 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}))
65difss2d 4108 . . . . . . . . 9 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ 𝒫 𝑋)
7 sspwuni 5013 . . . . . . . . 9 (ran 𝑓 ⊆ 𝒫 𝑋 ran 𝑓𝑋)
86, 7sylib 219 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓𝑋)
9 ssnum 9453 . . . . . . . 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 4427 . . . . . . . . 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 4639 . . . . . . . 8 ((ran 𝑓 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran 𝑓)
1412, 13sylib 219 . . . . . . 7 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ¬ ∅ ∈ ran 𝑓)
15 ac5num 9450 . . . . . . 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 772 . . . . . . 7 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦)) → 𝐴 ∈ V)
184ffnd 6508 . . . . . . . . . 10 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓 Fn 𝐴)
19 fveq2 6663 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑥) → (𝑦) = (‘(𝑓𝑥)))
20 id 22 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑥) → 𝑦 = (𝑓𝑥))
2119, 20eleq12d 2904 . . . . . . . . . . 11 (𝑦 = (𝑓𝑥) → ((𝑦) ∈ 𝑦 ↔ (‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2221ralrn 6846 . . . . . . . . . 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 477 . . . . . . . 8 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦) → ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥))
2524adantrl 712 . . . . . . 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 9462 . . . . . . 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 1927 . . . . 5 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
2928ralrimiva 3179 . . . 4 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
30 isacn 9458 . . . 4 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → (𝑋AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
3129, 30mpbird 258 . . 3 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → 𝑋AC 𝐴)
3231expcom 414 . 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 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wral 3135  Vcvv 3492  cdif 3930  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557   cuni 4830  dom cdm 5548  ran crn 5549   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  cardccrd 9352  AC wacn 9355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-card 9356  df-acn 9359
This theorem is referenced by:  acnnum  9466  fodomnum  9471  acacni  9554  dfac13  9556  iundom  9952  iunctb  9984
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