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Theorem elcnvlem 36823
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 5114 . 2 (𝐴𝐵 ↔ ∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
2 fveq2 5987 . . . . 5 (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹𝐴) = (𝐹‘⟨𝑢, 𝑣⟩))
3 vex 3080 . . . . . . 7 𝑢 ∈ V
4 vex 3080 . . . . . . 7 𝑣 ∈ V
53, 4opelvv 4982 . . . . . 6 𝑢, 𝑣⟩ ∈ (V × V)
63, 4op2ndd 6945 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → (2nd𝑥) = 𝑣)
73, 4op1std 6944 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → (1st𝑥) = 𝑢)
86, 7opeq12d 4246 . . . . . . 7 (𝑥 = ⟨𝑢, 𝑣⟩ → ⟨(2nd𝑥), (1st𝑥)⟩ = ⟨𝑣, 𝑢⟩)
9 elcnvlem.f . . . . . . 7 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2nd𝑥), (1st𝑥)⟩)
10 opex 4757 . . . . . . 7 𝑣, 𝑢⟩ ∈ V
118, 9, 10fvmpt 6075 . . . . . 6 (⟨𝑢, 𝑣⟩ ∈ (V × V) → (𝐹‘⟨𝑢, 𝑣⟩) = ⟨𝑣, 𝑢⟩)
125, 11ax-mp 5 . . . . 5 (𝐹‘⟨𝑢, 𝑣⟩) = ⟨𝑣, 𝑢
132, 12syl6eq 2564 . . . 4 (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹𝐴) = ⟨𝑣, 𝑢⟩)
1413eleq1d 2576 . . 3 (𝐴 = ⟨𝑢, 𝑣⟩ → ((𝐹𝐴) ∈ 𝐵 ↔ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
1514copsex2gb 5046 . 2 (∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ (𝐹𝐴) ∈ 𝐵))
161, 15bitri 262 1 (𝐴𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1938  Vcvv 3077  cop 4034  cmpt 4541   × cxp 4930  ccnv 4931  cfv 5689  1st c1st 6932  2nd c2nd 6933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fv 5697  df-1st 6934  df-2nd 6935
This theorem is referenced by:  elcnvintab  36824
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