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Theorem dmsnopg 5484
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 3112 . . . . . 6
2 vex 3112 . . . . . 6
31, 2opth1 4725 . . . . 5
43exlimiv 1722 . . . 4
5 opeq1 4217 . . . . 5
6 opeq2 4218 . . . . . . 7
76eqeq1d 2459 . . . . . 6
87spcegv 3195 . . . . 5
95, 8syl5 32 . . . 4
104, 9impbid2 204 . . 3
111eldm2 5206 . . . 4
12 opex 4716 . . . . . 6
1312elsnc 4053 . . . . 5
1413exbii 1667 . . . 4
1511, 14bitri 249 . . 3
16 elsn 4043 . . 3
1710, 15, 163bitr4g 288 . 2
1817eqrdv 2454 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  =wceq 1395  E.wex 1612  e.wcel 1818  {csn 4029  <.cop 4035  domcdm 5004
This theorem is referenced by:  dmsnopss  5485  dmpropg  5486  dmsnop  5487  rnsnopg  5492  fnsng  5640  funprg  5642  funtpg  5643  fntpg  5648  suppsnop  6932  funsnfsupp  7873  setsval  14656  eupap1  24976  estrreslem2  32644  bnj96  33923  bnj535  33948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-dm 5014
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