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Theorem ghgrplem1 21033
Description: Lemma for ghgrp 21035. (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 5679 . . 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 2667 . . 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 1623    e. wcel 1684   E.wrex 2544   -onto->wfo 5253   ` cfv 5255
This theorem is referenced by:  ghgrp  21035  ghablo  21036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263
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