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Theorem dmsnopg 4792
 Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dmsnopg

Proof of Theorem dmsnopg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . . . 6
2 vex 2560 . . . . . 6
31, 2opth1 3973 . . . . 5
43exlimiv 1489 . . . 4
5 opeq1 3549 . . . . 5
6 opeq2 3550 . . . . . . 7
76eqeq1d 2048 . . . . . 6
87spcegv 2641 . . . . 5
95, 8syl5 28 . . . 4
104, 9impbid2 131 . . 3
111eldm2 4533 . . . 4
121, 2opex 3966 . . . . . 6
1312elsn 3391 . . . . 5
1413exbii 1496 . . . 4
1511, 14bitri 173 . . 3
16 velsn 3392 . . 3
1710, 15, 163bitr4g 212 . 2
1817eqrdv 2038 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1243  wex 1381   wcel 1393  csn 3375  cop 3378   cdm 4345 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355 This theorem is referenced by:  dmpropg  4793  dmsnop  4794  rnsnopg  4799  elxp4  4808  fnsng  4947  funprg  4949  funtpg  4950  fntpg  4955
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