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Theorem fnopabco 31475
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 464 . . 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 4454 . . . . 5  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
53, 4eqtr4i 2434 . . . 4  |-  F  =  ( x  e.  A  |->  B )
65a1i 11 . . 3  |-  ( H  Fn  C  ->  F  =  ( x  e.  A  |->  B ) )
7 dffn5 5893 . . . 4  |-  ( H  Fn  C  <->  H  =  ( y  e.  C  |->  ( H `  y
) ) )
87biimpi 194 . . 3  |-  ( H  Fn  C  ->  H  =  ( y  e.  C  |->  ( H `  y ) ) )
9 fveq2 5848 . . 3  |-  ( y  =  B  ->  ( H `  y )  =  ( H `  B ) )
102, 6, 8, 9fmptco 6042 . 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 4454 . . 3  |-  ( x  e.  A  |->  ( H `
 B ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( H `  B ) ) }
1311, 12eqtr4i 2434 . 2  |-  G  =  ( x  e.  A  |->  ( H `  B
) )
1410, 13syl6reqr 2462 1  |-  ( H  Fn  C  ->  G  =  ( H  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   {copab 4451    |-> cmpt 4452    o. ccom 4826    Fn wfn 5563   ` cfv 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576
This theorem is referenced by:  opropabco  31476
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