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Theorem isprj1 25266
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 2809 . . 3  |-  ( P  e.  V  ->  P  e.  _V )
21adantr 451 . 2  |-  ( ( P  e.  V  /\  I  e.  W )  ->  P  e.  _V )
3 elex 2809 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
43adantl 452 . 2  |-  ( ( P  e.  V  /\  I  e.  W )  ->  I  e.  _V )
5 mptexg 5761 . . 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 4116 . . 3  |-  ( x  =  P  ->  (
f  e.  x  |->  ( f  |`  y )
)  =  ( f  e.  P  |->  ( f  |`  y ) ) )
8 reseq2 4966 . . . 4  |-  ( y  =  I  ->  (
f  |`  y )  =  ( f  |`  I ) )
98mpteq2dv 4123 . . 3  |-  ( y  =  I  ->  (
f  e.  P  |->  ( f  |`  y )
)  =  ( f  e.  P  |->  ( f  |`  I ) ) )
10 df-prj 25256 . . 3  |-  prj  =  ( x  e.  _V ,  y  e.  _V  |->  ( f  e.  x  |->  ( f  |`  y
) ) )
117, 9, 10ovmpt2g 5998 . 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 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093    |` cres 4707  (class class class)co 5874    prj cproj 25254
This theorem is referenced by:  isprj2  25267  prjmapcp2  25273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-prj 25256
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