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

Theorem cardval3ex 7046
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 6640 . . . 4 (𝑥𝐴 → (𝑥 ∈ V ∧ 𝐴 ∈ V))
21simprd 113 . . 3 (𝑥𝐴𝐴 ∈ V)
32rexlimivw 2545 . 2 (∃𝑥 ∈ On 𝑥𝐴𝐴 ∈ V)
4 breq1 3932 . . . 4 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
54cbvrexv 2655 . . 3 (∃𝑦 ∈ On 𝑦𝐴 ↔ ∃𝑥 ∈ On 𝑥𝐴)
6 intexrabim 4078 . . 3 (∃𝑦 ∈ On 𝑦𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
75, 6sylbir 134 . 2 (∃𝑥 ∈ On 𝑥𝐴 {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
8 breq2 3933 . . . . 5 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
98rabbidv 2675 . . . 4 (𝑧 = 𝐴 → {𝑦 ∈ On ∣ 𝑦𝑧} = {𝑦 ∈ On ∣ 𝑦𝐴})
109inteqd 3776 . . 3 (𝑧 = 𝐴 {𝑦 ∈ On ∣ 𝑦𝑧} = {𝑦 ∈ On ∣ 𝑦𝐴})
11 df-card 7041 . . 3 card = (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧})
1210, 11fvmptg 5497 . 2 ((𝐴 ∈ V ∧ {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V) → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
133, 7, 12syl2anc 408 1 (∃𝑥 ∈ On 𝑥𝐴 → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  wrex 2417  {crab 2420  Vcvv 2686   cint 3771   class class class wbr 3929  Oncon0 4285  cfv 5123  cen 6632  cardccrd 7040
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-en 6635  df-card 7041
This theorem is referenced by:  oncardval  7047  carden2bex  7050
  Copyright terms: Public domain W3C validator