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Theorem dfac8a 8736
 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 2610 . 2 recs((𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣)))) = recs((𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣))))
2 rneq 5272 . . . . 5 (𝑣 = 𝑓 → ran 𝑣 = ran 𝑓)
32difeq2d 3690 . . . 4 (𝑣 = 𝑓 → (𝐴 ∖ ran 𝑣) = (𝐴 ∖ ran 𝑓))
43fveq2d 6107 . . 3 (𝑣 = 𝑓 → (‘(𝐴 ∖ ran 𝑣)) = (‘(𝐴 ∖ ran 𝑓)))
54cbvmptv 4678 . 2 (𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣))) = (𝑓 ∈ V ↦ (‘(𝐴 ∖ ran 𝑓)))
61, 5dfac8alem 8735 1 (𝐴𝐵 → (∃𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  ‘cfv 5804  recscrecs 7354  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355  df-en 7842  df-card 8648 This theorem is referenced by:  ween  8741  acnnum  8758  dfac8  8840
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