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Theorem dfac8a 10024
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 2726 . 2 recs((𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣)))) = recs((𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣))))
2 rneq 5928 . . . . 5 (𝑣 = 𝑓 → ran 𝑣 = ran 𝑓)
32difeq2d 4117 . . . 4 (𝑣 = 𝑓 → (𝐴 ∖ ran 𝑣) = (𝐴 ∖ ran 𝑓))
43fveq2d 6888 . . 3 (𝑣 = 𝑓 → (‘(𝐴 ∖ ran 𝑣)) = (‘(𝐴 ∖ ran 𝑓)))
54cbvmptv 5254 . 2 (𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣))) = (𝑓 ∈ V ↦ (‘(𝐴 ∖ ran 𝑓)))
61, 5dfac8alem 10023 1 (𝐴𝐵 → (∃𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1773  wcel 2098  wne 2934  wral 3055  Vcvv 3468  cdif 3940  c0 4317  𝒫 cpw 4597  cmpt 5224  dom cdm 5669  ran crn 5670  cfv 6536  recscrecs 8368  cardccrd 9929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-en 8939  df-card 9933
This theorem is referenced by:  ween  10029  acnnum  10046  dfac8  10129
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