Theorem cnvoprab 8050

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 8044	. . 3 ⊢ {⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
3	2	cnveqi 5874	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		cnvopab 6138	. . 3 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)}
5		inopab 5829	. . 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 5550	. . . 4 ⊢ {⟨𝑧, 𝑎⟩ ∣ 𝜓} ⊆ {⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)}
8		sseqin2 4215	. . . 4 ⊢ ({⟨𝑧, 𝑎⟩ ∣ 𝜓} ⊆ {⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ↔ ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ 𝜓})
9	7, 8	mpbi 229	. . 3 ⊢ ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
10	4, 5, 9	3eqtr2i 2765	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
11	3, 10	eqtr3i 2761	1 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ∩ cin 3947   ⊆ wss 3948  ⟨cop 4634  {copab 5210   × cxp 5674  ^◡ccnv 5675  {coprab 7413

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-oprab 7416  df-1st 7979  df-2nd 7980

This theorem is referenced by:  f1od2  32381  dfxrn2  37713