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

Theorem resixp 6867
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 5010 . . 3  |-  ( F  e.  X_ x  e.  A  C  ->  ( F  |`  B )  e.  _V )
21adantl 452 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  e.  _V )
3 simpr 447 . . . . 5  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  F  e.  X_ x  e.  A  C )
4 elixp2 6836 . . . . 5  |-  ( F  e.  X_ x  e.  A  C 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  C ) )
53, 4sylib 188 . . . 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 968 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  F  Fn  A )
7 simpl 443 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  B  C_  A )
8 fnssres 5373 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
96, 7, 8syl2anc 642 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  Fn  B )
105simp3d 969 . . . 4  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  A  ( F `  x )  e.  C )
11 ssralv 3250 . . . 4  |-  ( B 
C_  A  ->  ( A. x  e.  A  ( F `  x )  e.  C  ->  A. x  e.  B  ( F `  x )  e.  C
) )
127, 10, 11sylc 56 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  B  ( F `  x )  e.  C )
13 fvres 5558 . . . . 5  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
1413eleq1d 2362 . . . 4  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  e.  C  <->  ( F `  x )  e.  C
) )
1514ralbiia 2588 . . 3  |-  ( A. x  e.  B  (
( F  |`  B ) `
 x )  e.  C  <->  A. x  e.  B  ( F `  x )  e.  C )
1612, 15sylibr 203 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  B  ( ( F  |`  B ) `  x
)  e.  C )
17 elixp2 6836 . 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 1136 1  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  e.  X_ x  e.  B  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165    |` cres 4707    Fn wfn 5266   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  resixpfo  6870  ixpfi2  7170  ptrescn  17349  ptuncnv  17514  ptcmplem2  17763  prjmapcp  25268  prl  25270
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-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-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-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ixp 6834
  Copyright terms: Public domain W3C validator