Theorem cnvopab 6086

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 2142, ax-12 2178. (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 6055	. 2 ⊢ Rel ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		relopabv 5764	. 2 ⊢ Rel {⟨𝑦, 𝑥⟩ ∣ 𝜑}
3		elopab 5470	. . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		excom 2163	. . . 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 5419	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦))
9	7, 6	opth 5419	. . . . . . 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 1849	. . . 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 5824	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ⟨𝑤, 𝑧⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
15		elopab 5470	. . 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 5733	1 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ⟨cop 4583  {copab 5154  ^◡ccnv 5618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-cnv 5627

This theorem is referenced by:  mptcnv  6088  cnvxp  6106  mptpreima  6187  f1ocnvd  7600  cnvoprab  7995  mapsncnv  8820  cnvepnep  9504  compsscnv  10265  dfiso2  17679  xkocnv  23699  lgsquadlem3  27291  axcontlem2  28914  cnvadj  31840  f1o3d  32577  xrninxp  38384  prjspeclsp  42605  fsovrfovd  44002