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Theorem elcnvlem 36926
 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 5222 . 2 (𝐴𝐵 ↔ ∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
2 fveq2 6103 . . . . 5 (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹𝐴) = (𝐹‘⟨𝑢, 𝑣⟩))
3 vex 3176 . . . . . . 7 𝑢 ∈ V
4 vex 3176 . . . . . . 7 𝑣 ∈ V
53, 4opelvv 5088 . . . . . 6 𝑢, 𝑣⟩ ∈ (V × V)
63, 4op2ndd 7070 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → (2nd𝑥) = 𝑣)
73, 4op1std 7069 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → (1st𝑥) = 𝑢)
86, 7opeq12d 4348 . . . . . . 7 (𝑥 = ⟨𝑢, 𝑣⟩ → ⟨(2nd𝑥), (1st𝑥)⟩ = ⟨𝑣, 𝑢⟩)
9 elcnvlem.f . . . . . . 7 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2nd𝑥), (1st𝑥)⟩)
10 opex 4859 . . . . . . 7 𝑣, 𝑢⟩ ∈ V
118, 9, 10fvmpt 6191 . . . . . 6 (⟨𝑢, 𝑣⟩ ∈ (V × V) → (𝐹‘⟨𝑢, 𝑣⟩) = ⟨𝑣, 𝑢⟩)
125, 11ax-mp 5 . . . . 5 (𝐹‘⟨𝑢, 𝑣⟩) = ⟨𝑣, 𝑢
132, 12syl6eq 2660 . . . 4 (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹𝐴) = ⟨𝑣, 𝑢⟩)
1413eleq1d 2672 . . 3 (𝐴 = ⟨𝑢, 𝑣⟩ → ((𝐹𝐴) ∈ 𝐵 ↔ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
1514copsex2gb 5153 . 2 (∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ (𝐹𝐴) ∈ 𝐵))
161, 15bitri 263 1 (𝐴𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹𝐴) ∈ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060 This theorem is referenced by:  elcnvintab  36927
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