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Theorem 1stval2 5813
 Description: Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
Assertion
Ref Expression
1stval2

Proof of Theorem 1stval2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 4428 . 2
2 vex 2605 . . . . . 6
3 vex 2605 . . . . . 6
42, 3op1st 5804 . . . . 5
52, 3op1stb 4235 . . . . 5
64, 5eqtr4i 2105 . . . 4
7 fveq2 5209 . . . 4
8 inteq 3647 . . . . 5
98inteqd 3649 . . . 4
106, 7, 93eqtr4a 2140 . . 3
1110exlimivv 1818 . 2
121, 11sylbi 119 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1285  wex 1422   wcel 1434  cvv 2602  cop 3409  cint 3644   cxp 4369  cfv 4932  c1st 5796 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fv 4940  df-1st 5798 This theorem is referenced by:  1stdm  5839
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