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Theorem eloprabi 6186
 Description: A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
eloprabi.1
eloprabi.2
eloprabi.3
Assertion
Ref Expression
eloprabi
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)

Proof of Theorem eloprabi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2289 . . . . . 6
21anbi1d 685 . . . . 5
323exbidv 1615 . . . 4
4 df-oprab 5862 . . . 4
53, 4elab2g 2916 . . 3
65ibi 232 . 2
7 opex 4237 . . . . . . . . . . 11
8 vex 2791 . . . . . . . . . . 11
97, 8op1std 6130 . . . . . . . . . 10
109fveq2d 5529 . . . . . . . . 9
11 vex 2791 . . . . . . . . . 10
12 vex 2791 . . . . . . . . . 10
1311, 12op1st 6128 . . . . . . . . 9
1410, 13syl6req 2332 . . . . . . . 8
15 eloprabi.1 . . . . . . . 8
1614, 15syl 15 . . . . . . 7
179fveq2d 5529 . . . . . . . . 9
1811, 12op2nd 6129 . . . . . . . . 9
1917, 18syl6req 2332 . . . . . . . 8
20 eloprabi.2 . . . . . . . 8
2119, 20syl 15 . . . . . . 7
227, 8op2ndd 6131 . . . . . . . . 9
2322eqcomd 2288 . . . . . . . 8
24 eloprabi.3 . . . . . . . 8
2523, 24syl 15 . . . . . . 7
2616, 21, 253bitrd 270 . . . . . 6
2726biimpa 470 . . . . 5
2827exlimiv 1666 . . . 4
2928exlimiv 1666 . . 3
3029exlimiv 1666 . 2
316, 30syl 15 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wex 1528   wceq 1623   wcel 1684  cop 3643  cfv 5255  coprab 5859  c1st 6120  c2nd 6121 This theorem is referenced by:  prismorcset2  25918  domcatfun  25925  codcatfun  25934 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-oprab 5862  df-1st 6122  df-2nd 6123
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