Theorem cnvoprab 7379

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 7373	. . 3 ⊢ {⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
3	2	cnveqi 5435	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		cnvopab 5674	. . 3 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)}
5		inopab 5391	. . 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 5136	. . . 4 ⊢ {⟨𝑧, 𝑎⟩ ∣ 𝜓} ⊆ {⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)}
8		sseqin2 3968	. . . 4 ⊢ ({⟨𝑧, 𝑎⟩ ∣ 𝜓} ⊆ {⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ↔ ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ 𝜓})
9	7, 8	mpbi 220	. . 3 ⊢ ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
10	4, 5, 9	3eqtr2i 2799	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
11	3, 10	eqtr3i 2795	1 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  Vcvv 3351   ∩ cin 3722   ⊆ wss 3723  ⟨cop 4322  {copab 4846   × cxp 5247  ^◡ccnv 5248  {coprab 6794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-oprab 6797  df-1st 7315  df-2nd 7316

This theorem is referenced by:  f1od2  29839  dfxrn2  34480