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Theorem oncardval 7065
 Description: The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
Assertion
Ref Expression
oncardval (𝐴 ∈ On → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Distinct variable group:   𝑥,𝐴

Proof of Theorem oncardval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 enrefg 6668 . . 3 (𝐴 ∈ On → 𝐴𝐴)
2 breq1 3941 . . . 4 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
32rspcev 2794 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
41, 3mpdan 418 . 2 (𝐴 ∈ On → ∃𝑦 ∈ On 𝑦𝐴)
5 cardval3ex 7064 . 2 (∃𝑦 ∈ On 𝑦𝐴 → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
64, 5syl 14 1 (𝐴 ∈ On → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  ∃wrex 2418  {crab 2421  ∩ cint 3780   class class class wbr 3938  Oncon0 4294  ‘cfv 5133   ≈ cen 6642  cardccrd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4555  df-rel 4556  df-cnv 4557  df-co 4558  df-dm 4559  df-rn 4560  df-res 4561  df-ima 4562  df-iota 5098  df-fun 5135  df-fn 5136  df-f 5137  df-f1 5138  df-fo 5139  df-f1o 5140  df-fv 5141  df-en 6645  df-card 7059 This theorem is referenced by:  cardonle  7066
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