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Theorem fnopabco 26415
Description: Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
fnopabco.1  |-  ( x  e.  A  ->  B  e.  C )
fnopabco.2  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
fnopabco.3  |-  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( H `  B ) ) }
Assertion
Ref Expression
fnopabco  |-  ( H  Fn  C  ->  G  =  ( H  o.  F ) )
Distinct variable groups:    x, C, y    y, B    x, H, y    x, A, y
Allowed substitution hints:    B( x)    F( x, y)    G( x, y)

Proof of Theorem fnopabco
StepHypRef Expression
1 fnopabco.1 . . . 4  |-  ( x  e.  A  ->  B  e.  C )
21adantl 453 . . 3  |-  ( ( H  Fn  C  /\  x  e.  A )  ->  B  e.  C )
3 fnopabco.2 . . . . 5  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
4 df-mpt 4260 . . . . 5  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
53, 4eqtr4i 2458 . . . 4  |-  F  =  ( x  e.  A  |->  B )
65a1i 11 . . 3  |-  ( H  Fn  C  ->  F  =  ( x  e.  A  |->  B ) )
7 dffn5 5764 . . . 4  |-  ( H  Fn  C  <->  H  =  ( y  e.  C  |->  ( H `  y
) ) )
87biimpi 187 . . 3  |-  ( H  Fn  C  ->  H  =  ( y  e.  C  |->  ( H `  y ) ) )
9 fveq2 5720 . . 3  |-  ( y  =  B  ->  ( H `  y )  =  ( H `  B ) )
102, 6, 8, 9fmptco 5893 . 2  |-  ( H  Fn  C  ->  ( H  o.  F )  =  ( x  e.  A  |->  ( H `  B ) ) )
11 fnopabco.3 . . 3  |-  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( H `  B ) ) }
12 df-mpt 4260 . . 3  |-  ( x  e.  A  |->  ( H `
 B ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( H `  B ) ) }
1311, 12eqtr4i 2458 . 2  |-  G  =  ( x  e.  A  |->  ( H `  B
) )
1410, 13syl6reqr 2486 1  |-  ( H  Fn  C  ->  G  =  ( H  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {copab 4257    e. cmpt 4258    o. ccom 4874    Fn wfn 5441   ` cfv 5446
This theorem is referenced by:  opropabco  26416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454
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