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Theorem resixp 6763
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 4968 . . 3  |-  ( F  e.  X_ x  e.  A  C  ->  ( F  |`  B )  e.  _V )
21adantl 277 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  e.  _V )
3 simpr 110 . . . . 5  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  F  e.  X_ x  e.  A  C )
4 elixp2 6732 . . . . 5  |-  ( F  e.  X_ x  e.  A  C 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  C ) )
53, 4sylib 122 . . . 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 1012 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  F  Fn  A )
7 simpl 109 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  B  C_  A )
8 fnssres 5351 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
96, 7, 8syl2anc 411 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  Fn  B )
105simp3d 1013 . . . 4  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  A  ( F `  x )  e.  C )
11 ssralv 3234 . . . 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 5561 . . . . 5  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
1413eleq1d 2258 . . . 4  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  e.  C  <->  ( F `  x )  e.  C
) )
1514ralbiia 2504 . . 3  |-  ( A. x  e.  B  (
( F  |`  B ) `
 x )  e.  C  <->  A. x  e.  B  ( F `  x )  e.  C )
1612, 15sylibr 134 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  B  ( ( F  |`  B ) `  x
)  e.  C )
17 elixp2 6732 . 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 1183 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 104    /\ w3a 980    e. wcel 2160   A.wral 2468   _Vcvv 2752    C_ wss 3144    |` cres 4649    Fn wfn 5233   ` cfv 5238   X_cixp 6728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-res 4659  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246  df-ixp 6729
This theorem is referenced by: (None)
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