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Theorem opabex2 6719
 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:   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem opabex2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 opabex2.1 . . 3
2 opabex2.2 . . 3
3 xpexg 6709 . . 3
41, 2, 3syl2anc 661 . 2
5 df-opab 4506 . . 3
6 simprl 755 . . . . . . . 8
7 opabex2.3 . . . . . . . . . 10
8 opabex2.4 . . . . . . . . . 10
9 opelxpi 5030 . . . . . . . . . 10
107, 8, 9syl2anc 661 . . . . . . . . 9
1110adantrl 715 . . . . . . . 8
126, 11eqeltrd 2555 . . . . . . 7
1312ex 434 . . . . . 6
1413exlimdvv 1701 . . . . 5
1514alrimiv 1695 . . . 4
16 abss 3569 . . . 4
1715, 16sylibr 212 . . 3
185, 17syl5eqss 3548 . 2
194, 18ssexd 4594 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369  wal 1377   wceq 1379  wex 1596   wcel 1767  cab 2452  cvv 3113   wss 3476  cop 4033  copab 4504   cxp 4997 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-opab 4506  df-xp 5005 This theorem is referenced by:  legval  23698
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