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Theorem dmsnopg 5341
 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 2959 . . . . . 6
2 vex 2959 . . . . . 6
31, 2opth1 4434 . . . . 5
43exlimiv 1644 . . . 4
5 opeq1 3984 . . . . 5
6 opeq2 3985 . . . . . . 7
76eqeq1d 2444 . . . . . 6
87spcegv 3037 . . . . 5
95, 8syl5 30 . . . 4
104, 9impbid2 196 . . 3
111eldm2 5068 . . . 4
12 opex 4427 . . . . . 6
1312elsnc 3837 . . . . 5
1413exbii 1592 . . . 4
1511, 14bitri 241 . . 3
16 elsn 3829 . . 3
1710, 15, 163bitr4g 280 . 2
1817eqrdv 2434 1
 Colors of variables: wff set class Syntax hints:   wi 4  wex 1550   wceq 1652   wcel 1725  csn 3814  cop 3817   cdm 4878 This theorem is referenced by:  dmsnopss  5342  dmpropg  5343  dmsnop  5344  rnsnopg  5349  fnsng  5498  funprg  5500  funtpg  5501  fntpg  5506  setsval  13493  eupap1  21698  bnj96  29236  bnj535  29261 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-dm 4888
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