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Theorem domrancur1clem 25201
Description: Lemma for domrancur1c 25202. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
domrancur1clem  |-  ( ( F  Fn  ( A  X.  B )  /\  ( A  e.  C  /\  B  e.  D
) )  ->  ( F  o.  `' ( 2nd  |`  M ) )  e.  _V )

Proof of Theorem domrancur1clem
StepHypRef Expression
1 xpexg 4800 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
2 fnex 5741 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  ( A  X.  B
)  e.  _V )  ->  F  e.  _V )
31, 2sylan2 460 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  ( A  e.  C  /\  B  e.  D
) )  ->  F  e.  _V )
4 fo2nd 6140 . . . . 5  |-  2nd : _V -onto-> _V
5 fofun 5452 . . . . 5  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
64, 5ax-mp 8 . . . 4  |-  Fun  2nd
7 funres 5293 . . . 4  |-  ( Fun 
2nd  ->  Fun  ( 2nd  |`  M ) )
86, 7ax-mp 8 . . 3  |-  Fun  ( 2nd  |`  M )
9 funcnvcnv 5308 . . 3  |-  ( Fun  ( 2nd  |`  M )  ->  Fun  `' `' ( 2nd  |`  M )
)
108, 9ax-mp 8 . 2  |-  Fun  `' `' ( 2nd  |`  M )
11 cofunex2g 5740 . 2  |-  ( ( F  e.  _V  /\  Fun  `' `' ( 2nd  |`  M ) )  ->  ( F  o.  `' ( 2nd  |`  M ) )  e.  _V )
123, 10, 11sylancl 643 1  |-  ( ( F  Fn  ( A  X.  B )  /\  ( A  e.  C  /\  B  e.  D
) )  ->  ( F  o.  `' ( 2nd  |`  M ) )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788    X. cxp 4687   `'ccnv 4688    |` cres 4691    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -onto->wfo 5253   2ndc2nd 6121
This theorem is referenced by:  valcurfn1  25204
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-2nd 6123
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