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Theorem cardval3ex 6910
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 6543 . . . 4 (𝑥𝐴 → (𝑥 ∈ V ∧ 𝐴 ∈ V))
21simprd 113 . . 3 (𝑥𝐴𝐴 ∈ V)
32rexlimivw 2498 . 2 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
4 breq1 3870 . . . 4 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
54cbvrexv 2605 . . 3 (∃𝑦 ∈ On 𝑦𝐴 ↔ ∃𝑥 ∈ On 𝑥𝐴)
6 intexrabim 4010 . . 3 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
75, 6sylbir 134 . 2 (∃𝑥 ∈ On 𝑥𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
8 breq2 3871 . . . . 5 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
98rabbidv 2622 . . . 4 (𝑧 = 𝐴 → {𝑦 ∈ On ∣ 𝑦𝑧} = {𝑦 ∈ On ∣ 𝑦𝐴})
109inteqd 3715 . . 3 (𝑧 = 𝐴 {𝑦 ∈ On ∣ 𝑦𝑧} = {𝑦 ∈ On ∣ 𝑦𝐴})
11 df-card 6905 . . 3 card = (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧})
1210, 11fvmptg 5415 . 2 ((𝐴 ∈ V ∧ {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V) → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
133, 7, 12syl2anc 404 1 (∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1296  wcel 1445  wrex 2371  {crab 2374  Vcvv 2633   cint 3710   class class class wbr 3867  Oncon0 4214  cfv 5049  cen 6535  cardccrd 6904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-en 6538  df-card 6905
This theorem is referenced by:  oncardval  6911  carden2bex  6914
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