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Theorem elcnvlem 40096
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 5721 . 2 (𝐴𝐵 ↔ ∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
2 fveq2 6643 . . . . 5 (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹𝐴) = (𝐹‘⟨𝑢, 𝑣⟩))
3 vex 3474 . . . . . . 7 𝑢 ∈ V
4 vex 3474 . . . . . . 7 𝑣 ∈ V
53, 4opelvv 5567 . . . . . 6 𝑢, 𝑣⟩ ∈ (V × V)
63, 4op2ndd 7675 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → (2nd𝑥) = 𝑣)
73, 4op1std 7674 . . . . . . . 8 (𝑥 = ⟨𝑢, 𝑣⟩ → (1st𝑥) = 𝑢)
86, 7opeq12d 4784 . . . . . . 7 (𝑥 = ⟨𝑢, 𝑣⟩ → ⟨(2nd𝑥), (1st𝑥)⟩ = ⟨𝑣, 𝑢⟩)
9 elcnvlem.f . . . . . . 7 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2nd𝑥), (1st𝑥)⟩)
10 opex 5329 . . . . . . 7 𝑣, 𝑢⟩ ∈ V
118, 9, 10fvmpt 6741 . . . . . 6 (⟨𝑢, 𝑣⟩ ∈ (V × V) → (𝐹‘⟨𝑢, 𝑣⟩) = ⟨𝑣, 𝑢⟩)
125, 11ax-mp 5 . . . . 5 (𝐹‘⟨𝑢, 𝑣⟩) = ⟨𝑣, 𝑢
132, 12syl6eq 2872 . . . 4 (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹𝐴) = ⟨𝑣, 𝑢⟩)
1413eleq1d 2896 . . 3 (𝐴 = ⟨𝑢, 𝑣⟩ → ((𝐹𝐴) ∈ 𝐵 ↔ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
1514copsex2gb 5652 . 2 (∃𝑢𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ (𝐹𝐴) ∈ 𝐵))
161, 15bitri 278 1 (𝐴𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2115  Vcvv 3471  cop 4546  cmpt 5119   × cxp 5526  ccnv 5527  cfv 6328  1st c1st 7662  2nd c2nd 7663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fv 6336  df-1st 7664  df-2nd 7665
This theorem is referenced by:  elcnvintab  40097
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