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Theorem resixp 7565
Description: Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
Assertion
Ref Expression
resixp  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  e.  X_ x  e.  B  C )
Distinct variable groups:    x, A    x, B    x, F
Allowed substitution hint:    C( x)

Proof of Theorem resixp
StepHypRef Expression
1 resexg 5167 . . 3  |-  ( F  e.  X_ x  e.  A  C  ->  ( F  |`  B )  e.  _V )
21adantl 467 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  e.  _V )
3 simpr 462 . . . . 5  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  F  e.  X_ x  e.  A  C )
4 elixp2 7534 . . . . 5  |-  ( F  e.  X_ x  e.  A  C 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  C ) )
53, 4sylib 199 . . . 4  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  C ) )
65simp2d 1018 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  F  Fn  A )
7 simpl 458 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  B  C_  A )
8 fnssres 5707 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
96, 7, 8syl2anc 665 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  Fn  B )
105simp3d 1019 . . . 4  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  A  ( F `  x )  e.  C )
11 ssralv 3531 . . . 4  |-  ( B 
C_  A  ->  ( A. x  e.  A  ( F `  x )  e.  C  ->  A. x  e.  B  ( F `  x )  e.  C
) )
127, 10, 11sylc 62 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  B  ( F `  x )  e.  C )
13 fvres 5895 . . . . 5  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
1413eleq1d 2498 . . . 4  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  e.  C  <->  ( F `  x )  e.  C
) )
1514ralbiia 2862 . . 3  |-  ( A. x  e.  B  (
( F  |`  B ) `
 x )  e.  C  <->  A. x  e.  B  ( F `  x )  e.  C )
1612, 15sylibr 215 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  B  ( ( F  |`  B ) `  x
)  e.  C )
17 elixp2 7534 . 2  |-  ( ( F  |`  B )  e.  X_ x  e.  B  C 
<->  ( ( F  |`  B )  e.  _V  /\  ( F  |`  B )  Fn  B  /\  A. x  e.  B  (
( F  |`  B ) `
 x )  e.  C ) )
182, 9, 16, 17syl3anbrc 1189 1  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  e.  X_ x  e.  B  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    e. wcel 1870   A.wral 2782   _Vcvv 3087    C_ wss 3442    |` cres 4856    Fn wfn 5596   ` cfv 5601   X_cixp 7530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-ixp 7531
This theorem is referenced by:  resixpfo  7568  ixpfi2  7878  ptrescn  20585  ptuncnv  20753  ptcmplem2  20999
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