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Theorem npincppr 25159
Description: A set of nuples is included in the cartesian product of the projections of the nuples. Bourbaki E.II.32. (Contributed by FL, 20-Jun-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
Hypothesis
Ref Expression
npincppr.1  |-  P  = 
X_ x  e.  A  B
Assertion
Ref Expression
npincppr  |-  ( ( F  C_  P  /\  P  e.  Q )  ->  F  C_  X_ x  e.  A  ( ( P  pr  x ) " F ) )
Distinct variable groups:    x, A    x, F    x, P    x, Q
Allowed substitution hint:    B( x)

Proof of Theorem npincppr
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ssel2 3175 . . . . . . 7  |-  ( ( F  C_  P  /\  f  e.  F )  ->  f  e.  P )
21adantlr 695 . . . . . 6  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  f  e.  F )  ->  f  e.  P )
3 npincppr.1 . . . . . 6  |-  P  = 
X_ x  e.  A  B
42, 3syl6eleq 2373 . . . . 5  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  f  e.  F )  ->  f  e.  X_ x  e.  A  B )
5 ixpf 6838 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f : A --> U_ x  e.  A  B
)
6 ffn 5389 . . . . 5  |-  ( f : A --> U_ x  e.  A  B  ->  f  Fn  A )
74, 5, 63syl 18 . . . 4  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  f  e.  F )  ->  f  Fn  A )
8 simplr 731 . . . . . . . 8  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  ->  P  e.  Q )
9 simprr 733 . . . . . . . 8  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  ->  x  e.  A )
102adantrr 697 . . . . . . . 8  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  -> 
f  e.  P )
11 valpr 25158 . . . . . . . 8  |-  ( ( P  e.  Q  /\  x  e.  A  /\  f  e.  P )  ->  ( ( P  pr  x ) `  f
)  =  ( f `
 x ) )
128, 9, 10, 11syl3anc 1182 . . . . . . 7  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  -> 
( ( P  pr  x ) `  f
)  =  ( f `
 x ) )
13 simprl 732 . . . . . . . 8  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  -> 
f  e.  F )
14 fvex 5539 . . . . . . . . . . . 12  |-  ( f `
 x )  e. 
_V
15 eqid 2283 . . . . . . . . . . . 12  |-  ( f  e.  P  |->  ( f `
 x ) )  =  ( f  e.  P  |->  ( f `  x ) )
1614, 15fnmpti 5372 . . . . . . . . . . 11  |-  ( f  e.  P  |->  ( f `
 x ) )  Fn  P
17 ispr1 25156 . . . . . . . . . . . . 13  |-  ( ( P  e.  Q  /\  x  e.  A )  ->  ( P  pr  x
)  =  ( f  e.  P  |->  ( f `
 x ) ) )
188, 9, 17syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  -> 
( P  pr  x
)  =  ( f  e.  P  |->  ( f `
 x ) ) )
1918fneq1d 5335 . . . . . . . . . . 11  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  -> 
( ( P  pr  x )  Fn  P  <->  ( f  e.  P  |->  ( f `  x ) )  Fn  P ) )
2016, 19mpbiri 224 . . . . . . . . . 10  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  -> 
( P  pr  x
)  Fn  P )
21 fnfun 5341 . . . . . . . . . 10  |-  ( ( P  pr  x )  Fn  P  ->  Fun  ( P  pr  x
) )
2220, 21syl 15 . . . . . . . . 9  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  ->  Fun  ( P  pr  x
) )
23 fndm 5343 . . . . . . . . . . 11  |-  ( ( P  pr  x )  Fn  P  ->  dom  ( P  pr  x
)  =  P )
2420, 23syl 15 . . . . . . . . . 10  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  ->  dom  ( P  pr  x
)  =  P )
2510, 24eleqtrrd 2360 . . . . . . . . 9  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  -> 
f  e.  dom  ( P  pr  x )
)
26 funfvima 5753 . . . . . . . . 9  |-  ( ( Fun  ( P  pr  x )  /\  f  e.  dom  ( P  pr  x ) )  -> 
( f  e.  F  ->  ( ( P  pr  x ) `  f
)  e.  ( ( P  pr  x )
" F ) ) )
2722, 25, 26syl2anc 642 . . . . . . . 8  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  -> 
( f  e.  F  ->  ( ( P  pr  x ) `  f
)  e.  ( ( P  pr  x )
" F ) ) )
2813, 27mpd 14 . . . . . . 7  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  -> 
( ( P  pr  x ) `  f
)  e.  ( ( P  pr  x )
" F ) )
2912, 28eqeltrrd 2358 . . . . . 6  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  ( f  e.  F  /\  x  e.  A ) )  -> 
( f `  x
)  e.  ( ( P  pr  x )
" F ) )
3029anassrs 629 . . . . 5  |-  ( ( ( ( F  C_  P  /\  P  e.  Q
)  /\  f  e.  F )  /\  x  e.  A )  ->  (
f `  x )  e.  ( ( P  pr  x ) " F
) )
3130ralrimiva 2626 . . . 4  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  f  e.  F )  ->  A. x  e.  A  ( f `  x )  e.  ( ( P  pr  x
) " F ) )
32 vex 2791 . . . . 5  |-  f  e. 
_V
3332elixp 6823 . . . 4  |-  ( f  e.  X_ x  e.  A  ( ( P  pr  x ) " F
)  <->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  ( ( P  pr  x
) " F ) ) )
347, 31, 33sylanbrc 645 . . 3  |-  ( ( ( F  C_  P  /\  P  e.  Q
)  /\  f  e.  F )  ->  f  e.  X_ x  e.  A  ( ( P  pr  x ) " F
) )
3534ex 423 . 2  |-  ( ( F  C_  P  /\  P  e.  Q )  ->  ( f  e.  F  ->  f  e.  X_ x  e.  A  ( ( P  pr  x ) " F ) ) )
3635ssrdv 3185 1  |-  ( ( F  C_  P  /\  P  e.  Q )  ->  F  C_  X_ x  e.  A  ( ( P  pr  x ) " F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   U_ciun 3905    e. cmpt 4077   dom cdm 4689   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   X_cixp 6817    pr cpro 25150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ixp 6818  df-pro 25152
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