Theorem cnvoprab 7758

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 7752	. . 3 ⊢ {⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
3	2	cnveqi 5745	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		cnvopab 5997	. . 3 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)}
5		inopab 5701	. . 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 5437	. . . 4 ⊢ {⟨𝑧, 𝑎⟩ ∣ 𝜓} ⊆ {⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)}
8		sseqin2 4192	. . . 4 ⊢ ({⟨𝑧, 𝑎⟩ ∣ 𝜓} ⊆ {⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ↔ ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ 𝜓})
9	7, 8	mpbi 232	. . 3 ⊢ ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
10	4, 5, 9	3eqtr2i 2850	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
11	3, 10	eqtr3i 2846	1 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  ⟨cop 4573  {copab 5128   × cxp 5553  ^◡ccnv 5554  {coprab 7157

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fv 6363  df-oprab 7160  df-1st 7689  df-2nd 7690

This theorem is referenced by:  f1od2  30457  dfxrn2  35643