Theorem cnvopab 6083

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 2144, ax-12 2180. (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 6052	. 2 ⊢ Rel ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		relopabv 5760	. 2 ⊢ Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑}
3		elopab 5465	. . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		excom 2165	. . . 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 3440	. . . . . . . 8 ⊢ 𝑤 ∈ V
7		vex 3440	. . . . . . . 8 ⊢ 𝑧 ∈ V
8	6, 7	opth 5414	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦))
9	7, 6	opth 5414	. . . . . . 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 624	. . . . 5 ⊢ ((⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
12	11	2exbii 1850	. . . 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 5820	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15		elopab 5465	. . 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 5729	1 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ⟨cop 4579  {copab 5151  ^◡ccnv 5613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622

This theorem is referenced by:  mptcnv  6085  cnvxp  6104  mptpreima  6185  f1ocnvd  7597  cnvoprab  7992  mapsncnv  8817  cnvepnep  9498  compsscnv  10262  dfiso2  17679  xkocnv  23729  lgsquadlem3  27320  axcontlem2  28943  cnvadj  31872  f1o3d  32608  xrninxp  38438  prjspeclsp  42704  fsovrfovd  44101