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Theorem mptcnv 4834
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 3904 . 2 (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶𝑥 = 𝐷)})
3 df-mpt 3901 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43cnveqi 4611 . . 3 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
5 cnvopab 4833 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
64, 5eqtri 2108 . 2 (𝑥𝐴𝐵) = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
7 df-mpt 3901 . 2 (𝑦𝐶𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶𝑥 = 𝐷)}
82, 6, 73eqtr4g 2145 1 (𝜑(𝑥𝐴𝐵) = (𝑦𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  {copab 3898  cmpt 3899  ccnv 4437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-mpt 3901  df-xp 4444  df-rel 4445  df-cnv 4446
This theorem is referenced by: (None)
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