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Theorem cardval 9123
Description: The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 8576 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 9051 . 2 (𝐴 ∈ V → 𝐴 ∈ dom card)
3 cardval3 8537 . 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 1474  wcel 1938  {crab 2804  Vcvv 3077   cint 4308   class class class wbr 4481  dom cdm 4932  Oncon0 5530  cfv 5689  cen 7714  cardccrd 8520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-ac2 9044
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-se 4892  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-isom 5698  df-riota 6388  df-wrecs 7169  df-recs 7231  df-en 7718  df-card 8524  df-ac 8698
This theorem is referenced by: (None)
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