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Theorem cnvopab 5978
Description: The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvopab {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem cnvopab
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5948 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 relopab 5677 . 2 Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑}
3 opelopabsbALT 5397 . . . 4 (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑)
4 sbcom2 2169 . . . 4 ([𝑧 / 𝑦][𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑)
53, 4bitri 278 . . 3 (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑)
6 vex 3482 . . . 4 𝑧 ∈ V
7 vex 3482 . . . 4 𝑤 ∈ V
86, 7opelcnv 5733 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
9 opelopabsbALT 5397 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑)
105, 8, 93bitr4i 306 . 2 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑})
111, 2, 10eqrelriiv 5644 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  [wsb 2070  wcel 2115  cop 4554  {copab 5109  ccnv 5535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-xp 5542  df-rel 5543  df-cnv 5544
This theorem is referenced by:  mptcnv  5979  cnvxp  5995  mptpreima  6073  f1ocnvd  7379  cnvoprab  7741  mapsncnv  8440  cnvepnep  9055  compsscnv  9778  dfiso2  17031  xkocnv  22408  lgsquadlem3  25955  axcontlem2  26748  cnvadj  29664  f1o3d  30369  cnvoprabOLD  30453  xrninxp  35700  prjspeclsp  39438  fsovrfovd  40543
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