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Theorem dfoprab4 6857
Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfoprab4.1
Assertion
Ref Expression
dfoprab4
Distinct variable groups:   , , ,   , , ,   , ,   ,   , , ,

Proof of Theorem dfoprab4
StepHypRef Expression
1 xpss 5114 . . . . . 6
21sseli 3499 . . . . 5
32adantr 465 . . . 4
43pm4.71ri 633 . . 3
54opabbii 4516 . 2
6 eleq1 2529 . . . . 5
7 opelxp 5034 . . . . 5
86, 7syl6bb 261 . . . 4
9 dfoprab4.1 . . . 4
108, 9anbi12d 710 . . 3
1110dfoprab3 6856 . 2
125, 11eqtri 2486 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  <.cop 4035  {copab 4509  X.cxp 5002  {coprab 6297
This theorem is referenced by:  dfoprab4f  6858  dfxp3  6860
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-eu 2286  df-mo 2287  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-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-oprab 6300  df-1st 6800  df-2nd 6801
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