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Theorem oncardval 6423
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 6274 . . 3 (𝐴 ∈ On → 𝐴𝐴)
2 breq1 3794 . . . 4 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
32rspcev 2673 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
41, 3mpdan 406 . 2 (𝐴 ∈ On → ∃𝑦 ∈ On 𝑦𝐴)
5 cardval3ex 6422 . 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 1259  wcel 1409  wrex 2324  {crab 2327   cint 3642   class class class wbr 3791  Oncon0 4127  cfv 4929  cen 6249  cardccrd 6416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-en 6252  df-card 6417
This theorem is referenced by:  cardonle  6424
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