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Theorem fnopabco 26108
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 4202 . . . . 5  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
53, 4eqtr4i 2403 . . . 4  |-  F  =  ( x  e.  A  |->  B )
65a1i 11 . . 3  |-  ( H  Fn  C  ->  F  =  ( x  e.  A  |->  B ) )
7 dffn5 5704 . . . 4  |-  ( H  Fn  C  <->  H  =  ( y  e.  C  |->  ( H `  y
) ) )
87biimpi 187 . . 3  |-  ( H  Fn  C  ->  H  =  ( y  e.  C  |->  ( H `  y ) ) )
9 fveq2 5661 . . 3  |-  ( y  =  B  ->  ( H `  y )  =  ( H `  B ) )
102, 6, 8, 9fmptco 5833 . 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 4202 . . 3  |-  ( x  e.  A  |->  ( H `
 B ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( H `  B ) ) }
1311, 12eqtr4i 2403 . 2  |-  G  =  ( x  e.  A  |->  ( H `  B
) )
1410, 13syl6reqr 2431 1  |-  ( H  Fn  C  ->  G  =  ( H  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {copab 4199    e. cmpt 4200    o. ccom 4815    Fn wfn 5382   ` cfv 5387
This theorem is referenced by:  opropabco  26109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395
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