Theorem cnvopab 6126

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 2177, ax-12 2214. (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 6095	. 2 ⊢ Rel ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		relopabv 5796	. 2 ⊢ Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑}
3		elopab 5499	. . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		excom 2198	. . . 4 ⊢ (∃𝑥∃𝑦(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑥(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		ancom 464	. . . . . . 7 ⊢ ((𝑤 = 𝑥 ∧ 𝑧 = 𝑦) ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥))
6		vex 3460	. . . . . . . 8 ⊢ 𝑤 ∈ V
7		vex 3460	. . . . . . . 8 ⊢ 𝑧 ∈ V
8	6, 7	opth 5446	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦))
9	7, 6	opth 5446	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ↔ (𝑧 = 𝑦 ∧ 𝑤 = 𝑥))
10	5, 8, 9	3bitr4i 305	. . . . . 6 ⊢ (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩)
11	10	anbi1i 633	. . . . 5 ⊢ ((⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
12	11	2exbii 1871	. . . 4 ⊢ (∃𝑦∃𝑥(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
13	3, 4, 12	3bitri 299	. . 3 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
14	7, 6	opelcnv 5855	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15		elopab 5499	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑} ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑦, 𝑥⟩ ∧ 𝜑))
16	13, 14, 15	3bitr4i 305	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑦, 𝑥⟩ ∣ 𝜑})
17	1, 2, 16	eqrelriiv 5764	1 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ⟨cop 4590  {copab 5164  ^◡ccnv 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657

This theorem is referenced by:  mptcnv  6127  cnvxp  6144  mptpreima  6227  f1ocnvd  7649  cnvoprab  8043  mapsncnv  8877  cnvepnep  9565  compsscnv  10330  dfiso2  17807  xkocnv  23876  lgsquadlem3  27448  axcontlem2  29168  cnvadj  32097  f1o3d  32830  vxp  38767  xrninxp  38919  prjspeclsp  43199  fsovrfovd  44590