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Theorem numacn 9062
 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 3352 . 2 (𝐴𝑉𝐴 ∈ V)
2 simpll 807 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → 𝑋 ∈ dom card)
3 elmapi 8045 . . . . . . . . . . . 12 (𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
43adantl 473 . . . . . . . . . . 11 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → 𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}))
5 frn 6214 . . . . . . . . . . 11 (𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}) → ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}))
64, 5syl 17 . . . . . . . . . 10 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}))
76difss2d 3883 . . . . . . . . 9 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → ran 𝑓 ⊆ 𝒫 𝑋)
8 sspwuni 4763 . . . . . . . . 9 (ran 𝑓 ⊆ 𝒫 𝑋 ran 𝑓𝑋)
97, 8sylib 208 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → ran 𝑓𝑋)
10 ssnum 9052 . . . . . . . 8 ((𝑋 ∈ dom card ∧ ran 𝑓𝑋) → ran 𝑓 ∈ dom card)
112, 9, 10syl2anc 696 . . . . . . 7 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → ran 𝑓 ∈ dom card)
12 ssdifin0 4194 . . . . . . . . 9 (ran 𝑓 ⊆ (𝒫 𝑋 ∖ {∅}) → (ran 𝑓 ∩ {∅}) = ∅)
136, 12syl 17 . . . . . . . 8 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → (ran 𝑓 ∩ {∅}) = ∅)
14 disjsn 4390 . . . . . . . 8 ((ran 𝑓 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ran 𝑓)
1513, 14sylib 208 . . . . . . 7 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → ¬ ∅ ∈ ran 𝑓)
16 ac5num 9049 . . . . . . 7 (( ran 𝑓 ∈ dom card ∧ ¬ ∅ ∈ ran 𝑓) → ∃(:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦))
1711, 15, 16syl2anc 696 . . . . . 6 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → ∃(:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦))
18 simpllr 817 . . . . . . 7 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) ∧ (:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦)) → 𝐴 ∈ V)
19 ffn 6206 . . . . . . . . . . 11 (𝑓:𝐴⟶(𝒫 𝑋 ∖ {∅}) → 𝑓 Fn 𝐴)
204, 19syl 17 . . . . . . . . . 10 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → 𝑓 Fn 𝐴)
21 fveq2 6352 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑥) → (𝑦) = (‘(𝑓𝑥)))
22 id 22 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑥) → 𝑦 = (𝑓𝑥))
2321, 22eleq12d 2833 . . . . . . . . . . 11 (𝑦 = (𝑓𝑥) → ((𝑦) ∈ 𝑦 ↔ (‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2423ralrn 6525 . . . . . . . . . 10 (𝑓 Fn 𝐴 → (∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦 ↔ ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2520, 24syl 17 . . . . . . . . 9 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → (∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦 ↔ ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2625biimpa 502 . . . . . . . 8 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦) → ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥))
2726adantrl 754 . . . . . . 7 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) ∧ (:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦)) → ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥))
28 acnlem 9061 . . . . . . 7 ((𝐴 ∈ V ∧ ∀𝑥𝐴 (‘(𝑓𝑥)) ∈ (𝑓𝑥)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
2918, 27, 28syl2anc 696 . . . . . 6 ((((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) ∧ (:ran 𝑓 ran 𝑓 ∧ ∀𝑦 ∈ ran 𝑓(𝑦) ∈ 𝑦)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
3017, 29exlimddv 2012 . . . . 5 (((𝑋 ∈ dom card ∧ 𝐴 ∈ V) ∧ 𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
3130ralrimiva 3104 . . . 4 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
32 isacn 9057 . . . 4 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → (𝑋AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
3331, 32mpbird 247 . . 3 ((𝑋 ∈ dom card ∧ 𝐴 ∈ V) → 𝑋AC 𝐴)
3433expcom 450 . 2 (𝐴 ∈ V → (𝑋 ∈ dom card → 𝑋AC 𝐴))
351, 34syl 17 1 (𝐴𝑉 → (𝑋 ∈ dom card → 𝑋AC 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632  ∃wex 1853   ∈ wcel 2139  ∀wral 3050  Vcvv 3340   ∖ cdif 3712   ∩ cin 3714   ⊆ wss 3715  ∅c0 4058  𝒫 cpw 4302  {csn 4321  ∪ cuni 4588  dom cdm 5266  ran crn 5267   Fn wfn 6044  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ↑𝑚 cmap 8023  cardccrd 8951  AC wacn 8954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-card 8955  df-acn 8958 This theorem is referenced by:  acnnum  9065  fodomnum  9070  acacni  9154  dfac13  9156  iundom  9556  iunctb  9588
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