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Theorem fnopabco 26491
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 452 . . 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 4095 . . . . 5  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
53, 4eqtr4i 2319 . . . 4  |-  F  =  ( x  e.  A  |->  B )
65a1i 10 . . 3  |-  ( H  Fn  C  ->  F  =  ( x  e.  A  |->  B ) )
7 dffn5 5584 . . . 4  |-  ( H  Fn  C  <->  H  =  ( y  e.  C  |->  ( H `  y
) ) )
87biimpi 186 . . 3  |-  ( H  Fn  C  ->  H  =  ( y  e.  C  |->  ( H `  y ) ) )
9 fveq2 5541 . . 3  |-  ( y  =  B  ->  ( H `  y )  =  ( H `  B ) )
102, 6, 8, 9fmptco 5707 . 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 4095 . . 3  |-  ( x  e.  A  |->  ( H `
 B ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( H `  B ) ) }
1311, 12eqtr4i 2319 . 2  |-  G  =  ( x  e.  A  |->  ( H `  B
) )
1410, 13syl6reqr 2347 1  |-  ( H  Fn  C  ->  G  =  ( H  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {copab 4092    e. cmpt 4093    o. ccom 4709    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  opropabco  26492
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  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-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-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-fv 5279
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