ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resixp Unicode version

Theorem resixp 6726
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 4942 . . 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 6695 . . . . 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 1010 . . 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 5324 . . 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 1011 . . . 4  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  A  ( F `  x )  e.  C )
11 ssralv 3219 . . . 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 5534 . . . . 5  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
1413eleq1d 2246 . . . 4  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  e.  C  <->  ( F `  x )  e.  C
) )
1514ralbiia 2491 . . 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 6695 . 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 1181 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 978    e. wcel 2148   A.wral 2455   _Vcvv 2737    C_ wss 3129    |` cres 4624    Fn wfn 5206   ` cfv 5211   X_cixp 6691
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-ixp 6692
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator