ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cardval3ex GIF version

Theorem cardval3ex 6716
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 6393 . . . 4 (𝑥𝐴 → (𝑥 ∈ V ∧ 𝐴 ∈ V))
21simprd 112 . . 3 (𝑥𝐴𝐴 ∈ V)
32rexlimivw 2479 . 2 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
4 breq1 3814 . . . 4 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
54cbvrexv 2584 . . 3 (∃𝑦 ∈ On 𝑦𝐴 ↔ ∃𝑥 ∈ On 𝑥𝐴)
6 intexrabim 3954 . . 3 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
75, 6sylbir 133 . 2 (∃𝑥 ∈ On 𝑥𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
8 breq2 3815 . . . . 5 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
98rabbidv 2601 . . . 4 (𝑧 = 𝐴 → {𝑦 ∈ On ∣ 𝑦𝑧} = {𝑦 ∈ On ∣ 𝑦𝐴})
109inteqd 3667 . . 3 (𝑧 = 𝐴 {𝑦 ∈ On ∣ 𝑦𝑧} = {𝑦 ∈ On ∣ 𝑦𝐴})
11 df-card 6711 . . 3 card = (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧})
1210, 11fvmptg 5325 . 2 ((𝐴 ∈ V ∧ {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V) → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
133, 7, 12syl2anc 403 1 (∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  wrex 2354  {crab 2357  Vcvv 2612   cint 3662   class class class wbr 3811  Oncon0 4154  cfv 4969  cen 6385  cardccrd 6710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fv 4977  df-en 6388  df-card 6711
This theorem is referenced by:  oncardval  6717  carden2bex  6720
  Copyright terms: Public domain W3C validator