Theorem cnvopab 6136

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 2175, ax-12 2212. (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 6105	. 2 ⊢ Rel ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		relopabv 5807	. 2 ⊢ Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑}
3		elopab 5510	. . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		excom 2196	. . . 4 ⊢ (∃𝑥∃𝑦(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑥(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		ancom 465	. . . . . . 7 ⊢ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥))
6		vex 3458	. . . . . . . 8 ⊢ 𝑤 ∈ V
7		vex 3458	. . . . . . . 8 ⊢ 𝑧 ∈ V
8	6, 7	opth 5457	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦))
9	7, 6	opth 5457	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥))
10	5, 8, 9	3bitr4i 306	. . . . . 6 ⊢ (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩)
11	10	anbi1i 635	. . . . 5 ⊢ ((⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
12	11	2exbii 1878	. . . 4 ⊢ (∃𝑦∃𝑥(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
13	3, 4, 12	3bitri 300	. . 3 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
14	7, 6	opelcnv 5866	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15		elopab 5510	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑} ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
16	13, 14, 15	3bitr4i 306	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑})
17	1, 2, 16	eqrelriiv 5775	1 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}

This proof depends on syntax axioms:   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142  ⟨cop 4594  {copab 5172  ^◡ccnv 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668

This theorem is used by:  mptcnv  6137  cnvxp  6153  mptpreima  6238  f1ocnvd  7663  cnvoprab  8055  mapsncnv  8889  cnvepnep  9575  compsscnv  10361  dfiso2  17835  xkocnv  23982  lgsquadlem3  27557  axcontlem2  29326  cnvadj  32255  f1o3d  32982  vxp  38940  xrninxp  39092  prjspeclsp  43372  fsovrfovd  44763