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Theorem reldmmpo 5848
 Description: The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypothesis
Ref Expression
rngop.1
Assertion
Ref Expression
reldmmpo
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   (,)   (,)   (,)

Proof of Theorem reldmmpo
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 reldmoprab 5822 . 2
2 rngop.1 . . . . 5
3 df-mpo 5745 . . . . 5
42, 3eqtri 2136 . . . 4
54dmeqi 4708 . . 3
65releqi 4590 . 2
71, 6mpbir 145 1
 Colors of variables: wff set class Syntax hints:   wa 103   wceq 1314   wcel 1463   cdm 4507   wrel 4512  coprab 5741   cmpo 5742 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-dm 4517  df-oprab 5744  df-mpo 5745 This theorem is referenced by:  reldmmap  6517  reldmsets  11894  reldmress  11923  reldmprds  12052
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