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Theorem cnvoprab 8084
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⟩ → (𝜓𝜑))
cnvoprab.2 (𝜓𝑎 ∈ (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
StepHypRef Expression
1 cnvoprab.1 . . . 4 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))
21dfoprab3 8078 . . 3 {⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
32cnveqi 5888 . 2 {⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4 cnvopab 6160 . . 3 {⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)}
5 inopab 5842 . . 3 ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)}
6 cnvoprab.2 . . . . 5 (𝜓𝑎 ∈ (V × V))
76ssopab2i 5560 . . . 4 {⟨𝑧, 𝑎⟩ ∣ 𝜓} ⊆ {⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)}
8 sseqin2 4231 . . . 4 ({⟨𝑧, 𝑎⟩ ∣ 𝜓} ⊆ {⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ↔ ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ 𝜓})
97, 8mpbi 230 . . 3 ({⟨𝑧, 𝑎⟩ ∣ 𝑎 ∈ (V × V)} ∩ {⟨𝑧, 𝑎⟩ ∣ 𝜓}) = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
104, 5, 93eqtr2i 2769 . 2 {⟨𝑎, 𝑧⟩ ∣ (𝑎 ∈ (V × V) ∧ 𝜓)} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
113, 10eqtr3i 2765 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963  cop 4637  {copab 5210   × cxp 5687  ccnv 5688  {coprab 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-oprab 7435  df-1st 8013  df-2nd 8014
This theorem is referenced by:  f1od2  32739  dfxrn2  38358
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