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Theorem cnvssrndm 5145
Description: The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
cnvssrndm 𝐴 ⊆ (ran 𝐴 × dom 𝐴)

Proof of Theorem cnvssrndm
StepHypRef Expression
1 relcnv 5001 . . 3 Rel 𝐴
2 relssdmrn 5144 . . 3 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
31, 2ax-mp 5 . 2 𝐴 ⊆ (dom 𝐴 × ran 𝐴)
4 df-rn 4633 . . 3 ran 𝐴 = dom 𝐴
5 dfdm4 4814 . . 3 dom 𝐴 = ran 𝐴
64, 5xpeq12i 4644 . 2 (ran 𝐴 × dom 𝐴) = (dom 𝐴 × ran 𝐴)
73, 6sseqtrri 3190 1 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
Colors of variables: wff set class
Syntax hints:  wss 3129   × cxp 4620  ccnv 4621  dom cdm 4622  ran crn 4623  Rel wrel 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-dm 4632  df-rn 4633
This theorem is referenced by: (None)
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