Theorem cnvoprab 8004

Description: The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) (Proof shortened by Thierry Arnoux, 20-Feb-2022.)

Hypotheses

Ref	Expression
cnvoprab.1	⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜑))
cnvoprab.2	⊢ (𝜓 → 𝑎 ∈ (V × V))

Assertion

Ref	Expression
cnvoprab	⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Distinct variable groups:   𝑥,𝑎,𝑦,𝑧   𝜑,𝑎   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑧,𝑎)

Proof of Theorem cnvoprab

Step	Hyp	Ref	Expression
1		cnvoprab.1	. . . 4 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜑))
2	1	dfoprab3 7998	. . 3 ⊢ {⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
3	2	cnveqi 5822	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		cnvopab 6093	. . 3 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)}
5		inopab 5777	. . 3 ⊢ ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)}
6		cnvoprab.2	. . . . 5 ⊢ (𝜓 → 𝑎 ∈ (V × V))
7	6	ssopab2i 5497	. . . 4 ⊢ {⟨𝑧, 𝑎⟩ ∣ 𝜓} ⊆ {⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)}
8		sseqin2 4174	. . . 4 ⊢ ({⟨𝑧, 𝑎⟩ ∣ 𝜓} ⊆ {⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ↔ ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ 𝜓})
9	7, 8	mpbi 230	. . 3 ⊢ ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
10	4, 5, 9	3eqtr2i 2764	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
11	3, 10	eqtr3i 2760	1 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3439   ∩ cin 3899   ⊆ wss 3900  ⟨cop 4585  {copab 5159   × cxp 5621  ^◡ccnv 5622  {coprab 7359

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-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680

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-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6447  df-fun 6493  df-fv 6499  df-oprab 7362  df-1st 7933  df-2nd 7934

This theorem is referenced by:  f1od2  32777  dfxrn2  38555