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Theorem cardval3ex 6322
Description: The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
cardval3ex (∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem cardval3ex
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 encv 6190 . . . 4 (𝑥𝐴 → (𝑥 ∈ V ∧ 𝐴 ∈ V))
21simprd 107 . . 3 (𝑥𝐴𝐴 ∈ V)
32rexlimivw 2426 . 2 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
4 breq1 3764 . . . 4 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
54cbvrexv 2531 . . 3 (∃𝑦 ∈ On 𝑦𝐴 ↔ ∃𝑥 ∈ On 𝑥𝐴)
6 intexrabim 3904 . . 3 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
75, 6sylbir 125 . 2 (∃𝑥 ∈ On 𝑥𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
8 breq2 3765 . . . . 5 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
98rabbidv 2546 . . . 4 (𝑧 = 𝐴 → {𝑦 ∈ On ∣ 𝑦𝑧} = {𝑦 ∈ On ∣ 𝑦𝐴})
109inteqd 3617 . . 3 (𝑧 = 𝐴 {𝑦 ∈ On ∣ 𝑦𝑧} = {𝑦 ∈ On ∣ 𝑦𝐴})
11 df-card 6317 . . 3 card = (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧})
1210, 11fvmptg 5211 . 2 ((𝐴 ∈ V ∧ {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V) → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
133, 7, 12syl2anc 391 1 (∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  wrex 2304  {crab 2307  Vcvv 2554   cint 3612   class class class wbr 3761  Oncon0 4072  cfv 4865  cen 6182  cardccrd 6316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-br 3762  df-opab 3816  df-mpt 3817  df-id 4027  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-iota 4830  df-fun 4867  df-fv 4873  df-en 6185  df-card 6317
This theorem is referenced by:  oncardval  6323  carden2bex  6326
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