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Theorem cardval 10586
Description: The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 10031 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypothesis
Ref Expression
cardval.1 𝐴 ∈ V
Assertion
Ref Expression
cardval (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem cardval
StepHypRef Expression
1 cardval.1 . 2 𝐴 ∈ V
2 numth3 10510 . 2 (𝐴 ∈ V → 𝐴 ∈ dom card)
3 cardval3 9992 . 2 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
41, 2, 3mp2b 10 1 (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480   cint 4946   class class class wbr 5143  dom cdm 5685  Oncon0 6384  cfv 6561  cen 8982  cardccrd 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-en 8986  df-card 9979  df-ac 10156
This theorem is referenced by: (None)
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