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Theorem oncardval 7008
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 6624 . . 3 (𝐴 ∈ On → 𝐴𝐴)
2 breq1 3900 . . . 4 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
32rspcev 2761 . . 3 ((𝐴 ∈ On ∧ 𝐴𝐴) → ∃𝑦 ∈ On 𝑦𝐴)
41, 3mpdan 415 . 2 (𝐴 ∈ On → ∃𝑦 ∈ On 𝑦𝐴)
5 cardval3ex 7007 . 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 1314  wcel 1463  wrex 2392  {crab 2395   cint 3739   class class class wbr 3897  Oncon0 4253  cfv 5091  cen 6598  cardccrd 7001
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-en 6601  df-card 7002
This theorem is referenced by:  cardonle  7009
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