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Theorem opabex2 6738
Description: Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)
Hypotheses
Ref Expression
opabex2.1
opabex2.2
opabex2.3
opabex2.4
Assertion
Ref Expression
opabex2
Distinct variable groups:   , ,   , ,   , ,

Proof of Theorem opabex2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 opabex2.1 . . 3
2 opabex2.2 . . 3
3 xpexg 6602 . . 3
41, 2, 3syl2anc 661 . 2
5 df-opab 4511 . . 3
6 simprl 756 . . . . . . 7
7 opabex2.3 . . . . . . . . 9
8 opabex2.4 . . . . . . . . 9
9 opelxpi 5036 . . . . . . . . 9
107, 8, 9syl2anc 661 . . . . . . . 8
1110adantrl 715 . . . . . . 7
126, 11eqeltrd 2545 . . . . . 6
1312ex 434 . . . . 5
1413exlimdvv 1725 . . . 4
1514abssdv 3573 . . 3
165, 15syl5eqss 3547 . 2
174, 16ssexd 4599 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442   cvv 3109  <.cop 4035  {copab 4509  X.cxp 5002
This theorem is referenced by:  legval  23971
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-8 1820  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-pow 4630  ax-pr 4691  ax-un 6592
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-ral 2812  df-rex 2813  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-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-opab 4511  df-xp 5010
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