MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghgrplem1 Unicode version

Theorem ghgrplem1 21049
Description: Lemma for ghgrp 21051. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
ghgrp.1  |-  ( ph  ->  F : X -onto-> Y
)
ghgrplem1.2  |-  ( (
ph  /\  w  e.  X )  ->  ps )
ghgrplem1.3  |-  ( C  =  ( F `  w )  ->  ( ch 
<->  ps ) )
Assertion
Ref Expression
ghgrplem1  |-  ( (
ph  /\  C  e.  Y )  ->  ch )
Distinct variable groups:    w, C    w, F    ph, w    ch, w    w, X    w, Y
Allowed substitution hint:    ps( w)

Proof of Theorem ghgrplem1
StepHypRef Expression
1 ghgrp.1 . . 3  |-  ( ph  ->  F : X -onto-> Y
)
2 foelrn 5695 . . 3  |-  ( ( F : X -onto-> Y  /\  C  e.  Y
)  ->  E. w  e.  X  C  =  ( F `  w ) )
31, 2sylan 457 . 2  |-  ( (
ph  /\  C  e.  Y )  ->  E. w  e.  X  C  =  ( F `  w ) )
4 ghgrplem1.2 . . . . 5  |-  ( (
ph  /\  w  e.  X )  ->  ps )
5 ghgrplem1.3 . . . . 5  |-  ( C  =  ( F `  w )  ->  ( ch 
<->  ps ) )
64, 5syl5ibrcom 213 . . . 4  |-  ( (
ph  /\  w  e.  X )  ->  ( C  =  ( F `  w )  ->  ch ) )
76rexlimdva 2680 . . 3  |-  ( ph  ->  ( E. w  e.  X  C  =  ( F `  w )  ->  ch ) )
87imp 418 . 2  |-  ( (
ph  /\  E. w  e.  X  C  =  ( F `  w ) )  ->  ch )
93, 8syldan 456 1  |-  ( (
ph  /\  C  e.  Y )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   -onto->wfo 5269   ` cfv 5271
This theorem is referenced by:  ghgrp  21051  ghablo  21052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279
  Copyright terms: Public domain W3C validator