Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfoprab4 Structured version   Unicode version

Theorem dfoprab4 6404
 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:   ,,,   ,,,   ,,   ,   ,,,
Allowed substitution hints:   (,)   (,,)   ()   ()

Proof of Theorem dfoprab4
StepHypRef Expression
1 xpss 4982 . . . . . 6
21sseli 3344 . . . . 5
32adantr 452 . . . 4
43pm4.71ri 615 . . 3
54opabbii 4272 . 2
6 eleq1 2496 . . . . 5
7 opelxp 4908 . . . . 5
86, 7syl6bb 253 . . . 4
9 dfoprab4.1 . . . 4
108, 9anbi12d 692 . . 3
1110dfoprab3 6403 . 2
125, 11eqtri 2456 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cvv 2956  cop 3817  copab 4265   cxp 4876  coprab 6082 This theorem is referenced by:  dfoprab4f  6405  dfxp3  6406 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-oprab 6085  df-1st 6349  df-2nd 6350
 Copyright terms: Public domain W3C validator