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Theorem prmapcp2 24569
Description: A projection is a mapping from a cartesian product to an element of the family implied in the product. Bourbaki E.II.34 cor. 1. (Contributed by FL, 19-Jun-2011.)
Hypotheses
Ref Expression
prmapcp2.1  |-  P  = 
X_ x  e.  A  B
prmapcp2.2  |-  ( x  =  I  ->  B  =  C )
Assertion
Ref Expression
prmapcp2  |-  ( ( P  e.  V  /\  I  e.  A )  ->  ( P  pr  I
) : P --> C )
Distinct variable groups:    x, A    x, C    x, I
Allowed substitution hints:    B( x)    P( x)    V( x)

Proof of Theorem prmapcp2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 prmapcp2.1 . . . . . 6  |-  P  = 
X_ x  e.  A  B
21eleq2i 2347 . . . . 5  |-  ( f  e.  P  <->  f  e.  X_ x  e.  A  B
)
32biimpi 186 . . . 4  |-  ( f  e.  P  ->  f  e.  X_ x  e.  A  B )
4 simpr 447 . . . 4  |-  ( ( P  e.  V  /\  I  e.  A )  ->  I  e.  A )
5 prmapcp2.2 . . . . 5  |-  ( x  =  I  ->  B  =  C )
65bclelnu 24567 . . . 4  |-  ( ( f  e.  X_ x  e.  A  B  /\  I  e.  A )  ->  ( f `  I
)  e.  C )
73, 4, 6syl2anr 464 . . 3  |-  ( ( ( P  e.  V  /\  I  e.  A
)  /\  f  e.  P )  ->  (
f `  I )  e.  C )
8 eqid 2283 . . 3  |-  ( f  e.  P  |->  ( f `
 I ) )  =  ( f  e.  P  |->  ( f `  I ) )
97, 8fmptd 5684 . 2  |-  ( ( P  e.  V  /\  I  e.  A )  ->  ( f  e.  P  |->  ( f `  I
) ) : P --> C )
10 ispr1 24568 . . 3  |-  ( ( P  e.  V  /\  I  e.  A )  ->  ( P  pr  I
)  =  ( f  e.  P  |->  ( f `
 I ) ) )
1110feq1d 5379 . 2  |-  ( ( P  e.  V  /\  I  e.  A )  ->  ( ( P  pr  I ) : P --> C 
<->  ( f  e.  P  |->  ( f `  I
) ) : P --> C ) )
129, 11mpbird 223 1  |-  ( ( P  e.  V  /\  I  e.  A )  ->  ( P  pr  I
) : P --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   X_cixp 6817    pr cpro 24562
This theorem is referenced by:  usptoplem  24958  istopx  24959  prcnt  24963
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 24564
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