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Theorem ghgrplem1 21954
Description: Lemma for ghgrp 21956. (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 5888 . . 3  |-  ( ( F : X -onto-> Y  /\  C  e.  Y
)  ->  E. w  e.  X  C  =  ( F `  w ) )
31, 2sylan 458 . 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 214 . . . 4  |-  ( (
ph  /\  w  e.  X )  ->  ( C  =  ( F `  w )  ->  ch ) )
76rexlimdva 2830 . . 3  |-  ( ph  ->  ( E. w  e.  X  C  =  ( F `  w )  ->  ch ) )
87imp 419 . 2  |-  ( (
ph  /\  E. w  e.  X  C  =  ( F `  w ) )  ->  ch )
93, 8syldan 457 1  |-  ( (
ph  /\  C  e.  Y )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   -onto->wfo 5452   ` cfv 5454
This theorem is referenced by:  ghgrp  21956  ghablo  21957
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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-fn 5457  df-f 5458  df-fo 5460  df-fv 5462
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