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Theorem fnopabco 28756
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 466 . . 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 4452 . . . . 5  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
53, 4eqtr4i 2483 . . . 4  |-  F  =  ( x  e.  A  |->  B )
65a1i 11 . . 3  |-  ( H  Fn  C  ->  F  =  ( x  e.  A  |->  B ) )
7 dffn5 5838 . . . 4  |-  ( H  Fn  C  <->  H  =  ( y  e.  C  |->  ( H `  y
) ) )
87biimpi 194 . . 3  |-  ( H  Fn  C  ->  H  =  ( y  e.  C  |->  ( H `  y ) ) )
9 fveq2 5791 . . 3  |-  ( y  =  B  ->  ( H `  y )  =  ( H `  B ) )
102, 6, 8, 9fmptco 5977 . 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 4452 . . 3  |-  ( x  e.  A  |->  ( H `
 B ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( H `  B ) ) }
1311, 12eqtr4i 2483 . 2  |-  G  =  ( x  e.  A  |->  ( H `  B
) )
1410, 13syl6reqr 2511 1  |-  ( H  Fn  C  ->  G  =  ( H  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {copab 4449    |-> cmpt 4450    o. ccom 4944    Fn wfn 5513   ` cfv 5518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526
This theorem is referenced by:  opropabco  28757
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