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Theorem isprj1 25163
Description: Definition of a projection.  I is a set of indices.  P is a cartesian product. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
isprj1  |-  ( ( P  e.  V  /\  I  e.  W )  ->  ( P  prj  I
)  =  ( f  e.  P  |->  ( f  |`  I ) ) )
Distinct variable groups:    f, I    P, f
Allowed substitution hints:    V( f)    W( f)

Proof of Theorem isprj1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . . 3  |-  ( P  e.  V  ->  P  e.  _V )
21adantr 451 . 2  |-  ( ( P  e.  V  /\  I  e.  W )  ->  P  e.  _V )
3 elex 2796 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
43adantl 452 . 2  |-  ( ( P  e.  V  /\  I  e.  W )  ->  I  e.  _V )
5 mptexg 5745 . . 3  |-  ( P  e.  V  ->  (
f  e.  P  |->  ( f  |`  I )
)  e.  _V )
65adantr 451 . 2  |-  ( ( P  e.  V  /\  I  e.  W )  ->  ( f  e.  P  |->  ( f  |`  I ) )  e.  _V )
7 mpteq1 4100 . . 3  |-  ( x  =  P  ->  (
f  e.  x  |->  ( f  |`  y )
)  =  ( f  e.  P  |->  ( f  |`  y ) ) )
8 reseq2 4950 . . . 4  |-  ( y  =  I  ->  (
f  |`  y )  =  ( f  |`  I ) )
98mpteq2dv 4107 . . 3  |-  ( y  =  I  ->  (
f  e.  P  |->  ( f  |`  y )
)  =  ( f  e.  P  |->  ( f  |`  I ) ) )
10 df-prj 25153 . . 3  |-  prj  =  ( x  e.  _V ,  y  e.  _V  |->  ( f  e.  x  |->  ( f  |`  y
) ) )
117, 9, 10ovmpt2g 5982 . 2  |-  ( ( P  e.  _V  /\  I  e.  _V  /\  (
f  e.  P  |->  ( f  |`  I )
)  e.  _V )  ->  ( P  prj  I
)  =  ( f  e.  P  |->  ( f  |`  I ) ) )
122, 4, 6, 11syl3anc 1182 1  |-  ( ( P  e.  V  /\  I  e.  W )  ->  ( P  prj  I
)  =  ( f  e.  P  |->  ( f  |`  I ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077    |` cres 4691  (class class class)co 5858    prj cproj 25151
This theorem is referenced by:  isprj2  25164  prjmapcp2  25170
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-prj 25153
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