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Theorem dfac8a 9774
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 2738 . 2 recs((𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣)))) = recs((𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣))))
2 rneq 5839 . . . . 5 (𝑣 = 𝑓 → ran 𝑣 = ran 𝑓)
32difeq2d 4057 . . . 4 (𝑣 = 𝑓 → (𝐴 ∖ ran 𝑣) = (𝐴 ∖ ran 𝑓))
43fveq2d 6771 . . 3 (𝑣 = 𝑓 → (‘(𝐴 ∖ ran 𝑣)) = (‘(𝐴 ∖ ran 𝑓)))
54cbvmptv 5187 . 2 (𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣))) = (𝑓 ∈ V ↦ (‘(𝐴 ∖ ran 𝑓)))
61, 5dfac8alem 9773 1 (𝐴𝐵 → (∃𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1782  wcel 2106  wne 2943  wral 3064  Vcvv 3430  cdif 3884  c0 4257  𝒫 cpw 4534  cmpt 5157  dom cdm 5585  ran crn 5586  cfv 6427  recscrecs 8189  cardccrd 9681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5485  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-we 5542  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-pred 6196  df-ord 6263  df-on 6264  df-suc 6266  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-ov 7271  df-2nd 7822  df-frecs 8085  df-wrecs 8116  df-recs 8190  df-en 8722  df-card 9685
This theorem is referenced by:  ween  9779  acnnum  9796  dfac8  9879
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