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Theorem cardval3ex 6423
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 6258 . . . 4 (𝑥𝐴 → (𝑥 ∈ V ∧ 𝐴 ∈ V))
21simprd 111 . . 3 (𝑥𝐴𝐴 ∈ V)
32rexlimivw 2446 . 2 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
4 breq1 3795 . . . 4 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
54cbvrexv 2551 . . 3 (∃𝑦 ∈ On 𝑦𝐴 ↔ ∃𝑥 ∈ On 𝑥𝐴)
6 intexrabim 3935 . . 3 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
75, 6sylbir 129 . 2 (∃𝑥 ∈ On 𝑥𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
8 breq2 3796 . . . . 5 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
98rabbidv 2566 . . . 4 (𝑧 = 𝐴 → {𝑦 ∈ On ∣ 𝑦𝑧} = {𝑦 ∈ On ∣ 𝑦𝐴})
109inteqd 3648 . . 3 (𝑧 = 𝐴 {𝑦 ∈ On ∣ 𝑦𝑧} = {𝑦 ∈ On ∣ 𝑦𝐴})
11 df-card 6418 . . 3 card = (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧})
1210, 11fvmptg 5276 . 2 ((𝐴 ∈ V ∧ {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V) → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
133, 7, 12syl2anc 397 1 (∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  wrex 2324  {crab 2327  Vcvv 2574   cint 3643   class class class wbr 3792  Oncon0 4128  cfv 4930  cen 6250  cardccrd 6417
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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
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 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-en 6253  df-card 6418
This theorem is referenced by:  oncardval  6424  carden2bex  6427
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