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Theorem brab2ga 5080
 Description: The law of concretion for a binary relation. See brab2a 5054 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
brab2ga.1
brab2ga.2
Assertion
Ref Expression
brab2ga
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,

Proof of Theorem brab2ga
StepHypRef Expression
1 brab2ga.2 . . . 4
2 opabssxp 5079 . . . 4
31, 2eqsstri 3533 . . 3
43brel 5053 . 2
5 df-br 4453 . . . 4
61eleq2i 2535 . . . 4
75, 6bitri 249 . . 3
8 brab2ga.1 . . . 4
98opelopab2a 4767 . . 3
107, 9syl5bb 257 . 2
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  <.cop 4035   class class class wbr 4452  {copab 4509  X.cxp 5002 This theorem is referenced by:  fnse  6917  ltxrlt  9676  ltxr  11353  gaorb  16345  ispgp  16612  efgcpbllema  16772  lmbr  19759  isphtpc  21494  vitalilem1  22017  vitalilem2  22018  vitalilem3  22019  filnetlem1  30196 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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 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-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-br 4453  df-opab 4511  df-xp 5010