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Theorem elcnvlem 43614
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 5888 . 2 (𝐴𝐵 ↔ ∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
2 fveq2 6906 . . . . 5 (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹𝐴) = (𝐹‘⟨𝑢, 𝑣⟩))
3 vex 3484 . . . . . . 7 𝑢 ∈ V
4 vex 3484 . . . . . . 7 𝑣 ∈ V
53, 4opelvv 5725 . . . . . 6 𝑢, 𝑣⟩ ∈ (V × V)
63, 4op2ndd 8025 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → (2nd𝑥) = 𝑣)
73, 4op1std 8024 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → (1st𝑥) = 𝑢)
86, 7opeq12d 4881 . . . . . . 7 (𝑥 = ⟨𝑢, 𝑣⟩ → ⟨(2nd𝑥), (1st𝑥)⟩ = ⟨𝑣, 𝑢⟩)
9 elcnvlem.f . . . . . . 7 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2nd𝑥), (1st𝑥)⟩)
10 opex 5469 . . . . . . 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 2793 . . . 4 (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹𝐴) = ⟨𝑣, 𝑢⟩)
1413eleq1d 2826 . . 3 (𝐴 = ⟨𝑢, 𝑣⟩ → ((𝐹𝐴) ∈ 𝐵 ↔ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
1514copsex2gb 5816 . 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 1540  wex 1779  wcel 2108  Vcvv 3480  cop 4632  cmpt 5225   × cxp 5683  ccnv 5684  cfv 6561  1st c1st 8012  2nd c2nd 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by:  elcnvintab  43615
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