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Theorem numacn 9692
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 3441 . 2 (𝐴𝑉𝐴 ∈ V)
2 simpll 767 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑋 ∈ dom card)
3 elmapi 8553 . . . . . . . . . . . 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 6574 . . . . . . . . . 10 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}))
65difss2d 4065 . . . . . . . . 9 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ran 𝑓 ⊆ 𝒫 𝑋)
7 sspwuni 5024 . . . . . . . . 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 9682 . . . . . . . 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 4413 . . . . . . . . 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 4643 . . . . . . . 8 ((ran 𝑓 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran 𝑓)
1412, 13sylib 221 . . . . . . 7 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ¬ ∅ ∈ ran 𝑓)
15 ac5num 9679 . . . . . . 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 776 . . . . . . 7 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦)) → 𝐴 ∈ V)
184ffnd 6567 . . . . . . . . . 10 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑓 Fn 𝐴)
19 fveq2 6738 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑥) → (𝑦) = (‘(𝑓𝑥)))
20 id 22 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑥) → 𝑦 = (𝑓𝑥))
2119, 20eleq12d 2834 . . . . . . . . . . 11 (𝑦 = (𝑓𝑥) → ((𝑦) ∈ 𝑦 ↔ (‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2221ralrn 6928 . . . . . . . . . 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 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 9691 . . . . . . 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 1943 . . . . 5 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
2928ralrimiva 3108 . . . 4 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
30 isacn 9687 . . . 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 1543  wex 1787  wcel 2112  wral 3064  Vcvv 3423  cdif 3880  cin 3882  wss 3883  c0 4253  𝒫 cpw 4529  {csn 4557   cuni 4835  dom cdm 5568  ran crn 5569   Fn wfn 6395  wf 6396  cfv 6400  (class class class)co 7234  m cmap 8531  cardccrd 9580  AC wacn 9583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-int 4876  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-se 5527  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-pred 6178  df-ord 6236  df-on 6237  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-isom 6409  df-riota 7191  df-ov 7237  df-oprab 7238  df-mpo 7239  df-1st 7782  df-2nd 7783  df-wrecs 8070  df-recs 8131  df-er 8414  df-map 8533  df-en 8650  df-dom 8651  df-card 9584  df-acn 9587
This theorem is referenced by:  acnnum  9695  fodomnum  9700  acacni  9783  dfac13  9785  iundom  10185  iunctb  10217
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