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Theorem fnopabco2b 25371
Description: Composition of a function with a function abstraction. Adapted from fnopabco 26388. (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
fnopabco2b.2  |-  F  =  ( x  e.  A  |->  B )
fnopabco2b.3  |-  G  =  ( x  e.  A  |->  ( H `  B
) )
Assertion
Ref Expression
fnopabco2b  |-  ( ( A. x  e.  A  B  e.  C  /\  H  Fn  C )  ->  G  =  ( H  o.  F ) )
Distinct variable groups:    x, A    x, C    x, H
Allowed substitution hints:    B( x)    F( x)    G( x)

Proof of Theorem fnopabco2b
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( A. x  e.  A  B  e.  C  /\  H  Fn  C )  ->  A. x  e.  A  B  e.  C )
2 fnopabco2b.2 . . . 4  |-  F  =  ( x  e.  A  |->  B )
32a1i 10 . . 3  |-  ( ( A. x  e.  A  B  e.  C  /\  H  Fn  C )  ->  F  =  ( x  e.  A  |->  B ) )
4 simpr 447 . . . 4  |-  ( ( A. x  e.  A  B  e.  C  /\  H  Fn  C )  ->  H  Fn  C )
5 dffn5 5568 . . . 4  |-  ( H  Fn  C  <->  H  =  ( y  e.  C  |->  ( H `  y
) ) )
64, 5sylib 188 . . 3  |-  ( ( A. x  e.  A  B  e.  C  /\  H  Fn  C )  ->  H  =  ( y  e.  C  |->  ( H `
 y ) ) )
7 fveq2 5525 . . 3  |-  ( y  =  B  ->  ( H `  y )  =  ( H `  B ) )
81, 3, 6, 7fmptcof 5692 . 2  |-  ( ( A. x  e.  A  B  e.  C  /\  H  Fn  C )  ->  ( H  o.  F
)  =  ( x  e.  A  |->  ( H `
 B ) ) )
9 fnopabco2b.3 . 2  |-  G  =  ( x  e.  A  |->  ( H `  B
) )
108, 9syl6reqr 2334 1  |-  ( ( A. x  e.  A  B  e.  C  /\  H  Fn  C )  ->  G  =  ( H  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077    o. ccom 4693    Fn wfn 5250   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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