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Theorem unidmrn 4773
Description: The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
Assertion
Ref Expression
unidmrn A = (dom A ∪ ran A)

Proof of Theorem unidmrn
StepHypRef Expression
1 relcnv 4626 . . . 4 Rel A
2 relfld 4769 . . . 4 (Rel A A = (dom A ∪ ran A))
31, 2ax-mp 7 . . 3 A = (dom A ∪ ran A)
43equncomi 3062 . 2 A = (ran A ∪ dom A)
5 dfdm4 4450 . . 3 dom A = ran A
6 df-rn 4279 . . 3 ran A = dom A
75, 6uneq12i 3068 . 2 (dom A ∪ ran A) = (ran A ∪ dom A)
84, 7eqtr4i 2041 1 A = (dom A ∪ ran A)
Colors of variables: wff set class
Syntax hints:   = wceq 1226  cun 2888   cuni 3550  ccnv 4267  dom cdm 4268  ran crn 4269  Rel wrel 4273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276  df-dm 4278  df-rn 4279
This theorem is referenced by:  relcnvfld  4774  dfdm2  4775
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