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

Theorem qsinxp 6983
 Description: Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Assertion
Ref Expression
qsinxp

Proof of Theorem qsinxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecinxp 6982 . . . . 5
21eqeq2d 2449 . . . 4
32rexbidva 2724 . . 3
43abbidv 2552 . 2
5 df-qs 6914 . 2
6 df-qs 6914 . 2
74, 5, 63eqtr4g 2495 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  cab 2424  wrex 2708   cin 3321   wss 3322   cxp 4879  cima 4884  cec 6906  cqs 6907 This theorem is referenced by:  pi1buni  19070  pi1bas3  19073 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-ec 6910  df-qs 6914
 Copyright terms: Public domain W3C validator