Theorem cnvopab 6104

Description: The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2147, ax-12 2185. (Revised by SN, 7-Jun-2025.)

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 6073	. 2 ⊢ Rel ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		relopabv 5780	. 2 ⊢ Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑}
3		elopab 5485	. . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		excom 2168	. . . 4 ⊢ (∃𝑥∃𝑦(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑥(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		ancom 460	. . . . . . 7 ⊢ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥))
6		vex 3446	. . . . . . . 8 ⊢ 𝑤 ∈ V
7		vex 3446	. . . . . . . 8 ⊢ 𝑧 ∈ V
8	6, 7	opth 5434	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦))
9	7, 6	opth 5434	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥))
10	5, 8, 9	3bitr4i 303	. . . . . 6 ⊢ (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩)
11	10	anbi1i 625	. . . . 5 ⊢ ((⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
12	11	2exbii 1851	. . . 4 ⊢ (∃𝑦∃𝑥(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
13	3, 4, 12	3bitri 297	. . 3 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
14	7, 6	opelcnv 5840	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15		elopab 5485	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑} ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
16	13, 14, 15	3bitr4i 303	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑})
17	1, 2, 16	eqrelriiv 5749	1 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ⟨cop 4588  {copab 5162  ^◡ccnv 5633

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641  df-cnv 5642

This theorem is referenced by:  mptcnv  6106  cnvxp  6125  mptpreima  6206  f1ocnvd  7621  cnvoprab  8016  mapsncnv  8845  cnvepnep  9531  compsscnv  10295  dfiso2  17710  xkocnv  23775  lgsquadlem3  27366  axcontlem2  29056  cnvadj  31986  f1o3d  32722  vxp  38543  xrninxp  38695  prjspeclsp  42999  fsovrfovd  44394