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Theorem resixp 7290
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 5144 . . 3  |-  ( F  e.  X_ x  e.  A  C  ->  ( F  |`  B )  e.  _V )
21adantl 466 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  e.  _V )
3 simpr 461 . . . . 5  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  F  e.  X_ x  e.  A  C )
4 elixp2 7259 . . . . 5  |-  ( F  e.  X_ x  e.  A  C 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  C ) )
53, 4sylib 196 . . . 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 1001 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  F  Fn  A )
7 simpl 457 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  B  C_  A )
8 fnssres 5519 . . 3  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
96, 7, 8syl2anc 661 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  -> 
( F  |`  B )  Fn  B )
105simp3d 1002 . . . 4  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  A  ( F `  x )  e.  C )
11 ssralv 3411 . . . 4  |-  ( B 
C_  A  ->  ( A. x  e.  A  ( F `  x )  e.  C  ->  A. x  e.  B  ( F `  x )  e.  C
) )
127, 10, 11sylc 60 . . 3  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  B  ( F `  x )  e.  C )
13 fvres 5699 . . . . 5  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
1413eleq1d 2504 . . . 4  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  e.  C  <->  ( F `  x )  e.  C
) )
1514ralbiia 2742 . . 3  |-  ( A. x  e.  B  (
( F  |`  B ) `
 x )  e.  C  <->  A. x  e.  B  ( F `  x )  e.  C )
1612, 15sylibr 212 . 2  |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  A. x  e.  B  ( ( F  |`  B ) `  x
)  e.  C )
17 elixp2 7259 . 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 1172 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 369    /\ w3a 965    e. wcel 1756   A.wral 2710   _Vcvv 2967    C_ wss 3323    |` cres 4837    Fn wfn 5408   ` cfv 5413   X_cixp 7255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-res 4847  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421  df-ixp 7256
This theorem is referenced by:  resixpfo  7293  ixpfi2  7601  ptrescn  19187  ptuncnv  19355  ptcmplem2  19600
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