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Theorem numacn 9469
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 3498 . 2 (𝐴𝑉𝐴 ∈ V)
2 simpll 766 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑋 ∈ dom card)
3 elmapi 8420 . . . . . . . . . . . 12 (𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
43adantl 485 . . . . . . . . . . 11 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
54frnd 6510 . . . . . . . . . 10 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}))
65difss2d 4097 . . . . . . . . 9 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ 𝒫 𝑋)
7 sspwuni 5009 . . . . . . . . 9 (ran 𝑓 ⊆ 𝒫 𝑋 ran 𝑓𝑋)
86, 7sylib 221 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓𝑋)
9 ssnum 9459 . . . . . . . 8 ((𝑋 ∈ dom card ∧ ran 𝑓𝑋) → ran 𝑓 ∈ dom card)
102, 8, 9syl2anc 587 . . . . . . 7 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ∈ dom card)
11 ssdifin0 4414 . . . . . . . . 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 4632 . . . . . . . 8 ((ran 𝑓 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran 𝑓)
1412, 13sylib 221 . . . . . . 7 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ¬ ∅ ∈ ran 𝑓)
15 ac5num 9456 . . . . . . 7 (( ran 𝑓 ∈ dom card ∧ ¬ ∅ ∈ ran 𝑓) → ∃(:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦))
1610, 14, 15syl2anc 587 . . . . . 6 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃(:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦))
17 simpllr 775 . . . . . . 7 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦)) → 𝐴 ∈ V)
184ffnd 6504 . . . . . . . . . 10 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓 Fn 𝐴)
19 fveq2 6659 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑥) → (𝑦) = (‘(𝑓𝑥)))
20 id 22 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑥) → 𝑦 = (𝑓𝑥))
2119, 20eleq12d 2910 . . . . . . . . . . 11 (𝑦 = (𝑓𝑥) → ((𝑦) ∈ 𝑦 ↔ (‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2221ralrn 6843 . . . . . . . . . 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 480 . . . . . . . 8 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦) → ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥))
2524adantrl 715 . . . . . . 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 9468 . . . . . . 7 ((𝐴 ∈ V ∧ ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
2717, 25, 26syl2anc 587 . . . . . 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 1937 . . . . 5 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
2928ralrimiva 3177 . . . 4 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
30 isacn 9464 . . . 4 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → (𝑋AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
3129, 30mpbird 260 . . 3 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → 𝑋AC 𝐴)
3231expcom 417 . 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 209  wa 399   = wceq 1538  wex 1781  wcel 2115  wral 3133  Vcvv 3480  cdif 3916  cin 3918  wss 3919  c0 4276  𝒫 cpw 4522  {csn 4550   cuni 4825  dom cdm 5543  ran crn 5544   Fn wfn 6339  wf 6340  cfv 6344  (class class class)co 7146  m cmap 8398  cardccrd 9357  AC wacn 9360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-isom 6353  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-card 9361  df-acn 9364
This theorem is referenced by:  acnnum  9472  fodomnum  9477  acacni  9560  dfac13  9562  iundom  9958  iunctb  9990
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