Theorem cnvopab 6096

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.

Step	Hyp	Ref	Expression
1		relcnv 6061	. 2 ⊢ Rel ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		relopabv 5782	. 2 ⊢ Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑}
3		vopelopabsb 5491	. . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑥][𝑧 / 𝑦]𝜑)
4		sbcom2 2161	. . . 4 ⊢ ([𝑤 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑)
5	3, 4	bitri 274	. . 3 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑)
6		vex 3450	. . . 4 ⊢ 𝑧 ∈ V
7		vex 3450	. . . 4 ⊢ 𝑤 ∈ V
8	6, 7	opelcnv 5842	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
9		vopelopabsb 5491	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑} ↔ [𝑧 / 𝑦][𝑤 / 𝑥]𝜑)
10	5, 8, 9	3bitr4i 302	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑})
11	1, 2, 10	eqrelriiv 5751	1 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}

Syntax hints:   = wceq 1541  [wsb 2067   ∈ wcel 2106  ⟨cop 4597  {copab 5172  ^◡ccnv 5637

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646

This theorem is referenced by:  mptcnv  6097  cnvxp  6114  mptpreima  6195  f1ocnvd  7609  cnvoprab  7997  mapsncnv  8838  cnvepnep  9553  compsscnv  10316  dfiso2  17669  xkocnv  23202  lgsquadlem3  26767  axcontlem2  27977  cnvadj  30897  f1o3d  31608  cnvoprabOLD  31705  xrninxp  36927  prjspeclsp  41008  fsovrfovd  42403