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Theorem dfac8a 10071
Description: Numeration theorem: every set with a choice function on its power set is numerable. With AC, this reduces to the statement that every set is numerable. Similar to Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)
Assertion
Ref Expression
dfac8a (𝐴𝐵 → (∃𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Distinct variable groups:   𝑦,,𝐴   𝐵,
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem dfac8a
Dummy variables 𝑓 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . 2 recs((𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣)))) = recs((𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣))))
2 rneq 5946 . . . . 5 (𝑣 = 𝑓 → ran 𝑣 = ran 𝑓)
32difeq2d 4125 . . . 4 (𝑣 = 𝑓 → (𝐴 ∖ ran 𝑣) = (𝐴 ∖ ran 𝑓))
43fveq2d 6909 . . 3 (𝑣 = 𝑓 → (‘(𝐴 ∖ ran 𝑣)) = (‘(𝐴 ∖ ran 𝑓)))
54cbvmptv 5254 . 2 (𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣))) = (𝑓 ∈ V ↦ (‘(𝐴 ∖ ran 𝑓)))
61, 5dfac8alem 10070 1 (𝐴𝐵 → (∃𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1778  wcel 2107  wne 2939  wral 3060  Vcvv 3479  cdif 3947  c0 4332  𝒫 cpw 4599  cmpt 5224  dom cdm 5684  ran crn 5685  cfv 6560  recscrecs 8411  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-en 8987  df-card 9980
This theorem is referenced by:  ween  10076  acnnum  10093  dfac8  10177
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