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Theorem dfac8a 10054
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 2728 . 2 recs((𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣)))) = recs((𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣))))
2 rneq 5938 . . . . 5 (𝑣 = 𝑓 → ran 𝑣 = ran 𝑓)
32difeq2d 4120 . . . 4 (𝑣 = 𝑓 → (𝐴 ∖ ran 𝑣) = (𝐴 ∖ ran 𝑓))
43fveq2d 6901 . . 3 (𝑣 = 𝑓 → (‘(𝐴 ∖ ran 𝑣)) = (‘(𝐴 ∖ ran 𝑓)))
54cbvmptv 5261 . 2 (𝑣 ∈ V ↦ (‘(𝐴 ∖ ran 𝑣))) = (𝑓 ∈ V ↦ (‘(𝐴 ∖ ran 𝑓)))
61, 5dfac8alem 10053 1 (𝐴𝐵 → (∃𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1774  wcel 2099  wne 2937  wral 3058  Vcvv 3471  cdif 3944  c0 4323  𝒫 cpw 4603  cmpt 5231  dom cdm 5678  ran crn 5679  cfv 6548  recscrecs 8391  cardccrd 9959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-en 8965  df-card 9963
This theorem is referenced by:  ween  10059  acnnum  10076  dfac8  10159
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