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Theorem mptcnv 5987
 Description: The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)
Hypothesis
Ref Expression
mptcnv.1 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑦𝐶𝑥 = 𝐷)))
Assertion
Ref Expression
mptcnv (𝜑(𝑥𝐴𝐵) = (𝑦𝐶𝐷))
Distinct variable groups:   𝑥,𝑦,𝜑   𝑥,𝐶   𝑥,𝐷   𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑦)

Proof of Theorem mptcnv
StepHypRef Expression
1 mptcnv.1 . . 3 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑦𝐶𝑥 = 𝐷)))
21opabbidv 5119 . 2 (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶𝑥 = 𝐷)})
3 df-mpt 5134 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43cnveqi 5733 . . 3 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
5 cnvopab 5986 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
64, 5eqtri 2847 . 2 (𝑥𝐴𝐵) = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
7 df-mpt 5134 . 2 (𝑦𝐶𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶𝑥 = 𝐷)}
82, 6, 73eqtr4g 2884 1 (𝜑(𝑥𝐴𝐵) = (𝑦𝐶𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  {copab 5115   ↦ cmpt 5133  ◡ccnv 5542 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 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 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 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-mpt 5134  df-xp 5549  df-rel 5550  df-cnv 5551 This theorem is referenced by:  nvocnv  7032  mptfzshft  15135  fprodrev  15333  pt1hmeo  22420  ballotlemrinv  31876  dssmapnvod  40666
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