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Theorem elcnvlem 43591
Description: Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.)
Hypothesis
Ref Expression
elcnvlem.f 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2nd𝑥), (1st𝑥)⟩)
Assertion
Ref Expression
elcnvlem (𝐴𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹𝐴) ∈ 𝐵))

Proof of Theorem elcnvlem
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elcnv2 5891 . 2 (𝐴𝐵 ↔ ∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
2 fveq2 6907 . . . . 5 (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹𝐴) = (𝐹‘⟨𝑢, 𝑣⟩))
3 vex 3482 . . . . . . 7 𝑢 ∈ V
4 vex 3482 . . . . . . 7 𝑣 ∈ V
53, 4opelvv 5729 . . . . . 6 𝑢, 𝑣⟩ ∈ (V × V)
63, 4op2ndd 8024 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → (2nd𝑥) = 𝑣)
73, 4op1std 8023 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → (1st𝑥) = 𝑢)
86, 7opeq12d 4886 . . . . . . 7 (𝑥 = ⟨𝑢, 𝑣⟩ → ⟨(2nd𝑥), (1st𝑥)⟩ = ⟨𝑣, 𝑢⟩)
9 elcnvlem.f . . . . . . 7 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2nd𝑥), (1st𝑥)⟩)
10 opex 5475 . . . . . . 7 𝑣, 𝑢⟩ ∈ V
118, 9, 10fvmpt 7016 . . . . . 6 (⟨𝑢, 𝑣⟩ ∈ (V × V) → (𝐹‘⟨𝑢, 𝑣⟩) = ⟨𝑣, 𝑢⟩)
125, 11ax-mp 5 . . . . 5 (𝐹‘⟨𝑢, 𝑣⟩) = ⟨𝑣, 𝑢
132, 12eqtrdi 2791 . . . 4 (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹𝐴) = ⟨𝑣, 𝑢⟩)
1413eleq1d 2824 . . 3 (𝐴 = ⟨𝑢, 𝑣⟩ → ((𝐹𝐴) ∈ 𝐵 ↔ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
1514copsex2gb 5819 . 2 (∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ (𝐹𝐴) ∈ 𝐵))
161, 15bitri 275 1 (𝐴𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cop 4637  cmpt 5231   × cxp 5687  ccnv 5688  cfv 6563  1st c1st 8011  2nd c2nd 8012
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-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  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-1st 8013  df-2nd 8014
This theorem is referenced by:  elcnvintab  43592
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