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Theorem offveqb 6258
Description: Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
offveq.1  |-  ( ph  ->  A  e.  V )
offveq.2  |-  ( ph  ->  F  Fn  A )
offveq.3  |-  ( ph  ->  G  Fn  A )
offveq.4  |-  ( ph  ->  H  Fn  A )
offveq.5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
offveq.6  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  C )
Assertion
Ref Expression
offveqb  |-  ( ph  ->  ( H  =  ( F  o F R G )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
Distinct variable groups:    x, A    x, F    x, G    x, H    ph, x    x, R
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem offveqb
StepHypRef Expression
1 offveq.4 . . . 4  |-  ( ph  ->  H  Fn  A )
2 dffn5 5704 . . . 4  |-  ( H  Fn  A  <->  H  =  ( x  e.  A  |->  ( H `  x
) ) )
31, 2sylib 189 . . 3  |-  ( ph  ->  H  =  ( x  e.  A  |->  ( H `
 x ) ) )
4 offveq.2 . . . 4  |-  ( ph  ->  F  Fn  A )
5 offveq.3 . . . 4  |-  ( ph  ->  G  Fn  A )
6 offveq.1 . . . 4  |-  ( ph  ->  A  e.  V )
7 inidm 3486 . . . 4  |-  ( A  i^i  A )  =  A
8 offveq.5 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
9 offveq.6 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  =  C )
104, 5, 6, 6, 7, 8, 9offval 6244 . . 3  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  A  |->  ( B R C ) ) )
113, 10eqeq12d 2394 . 2  |-  ( ph  ->  ( H  =  ( F  o F R G )  <->  ( x  e.  A  |->  ( H `
 x ) )  =  ( x  e.  A  |->  ( B R C ) ) ) )
12 fvex 5675 . . . . 5  |-  ( H `
 x )  e. 
_V
1312a1i 11 . . . 4  |-  ( ph  ->  ( H `  x
)  e.  _V )
1413ralrimivw 2726 . . 3  |-  ( ph  ->  A. x  e.  A  ( H `  x )  e.  _V )
15 mpteqb 5751 . . 3  |-  ( A. x  e.  A  ( H `  x )  e.  _V  ->  ( (
x  e.  A  |->  ( H `  x ) )  =  ( x  e.  A  |->  ( B R C ) )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
1614, 15syl 16 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  ( H `  x ) )  =  ( x  e.  A  |->  ( B R C ) )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
1711, 16bitrd 245 1  |-  ( ph  ->  ( H  =  ( F  o F R G )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   _Vcvv 2892    e. cmpt 4200    Fn wfn 5382   ` cfv 5387  (class class class)co 6013    o Fcof 6235
This theorem is referenced by:  eqlkr2  29266
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-rep 4254  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-reu 2649  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-iun 4030  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-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237
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