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Theorem offveqb 6054
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  oF 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 dffn5im 5517 . . . 4  |-  ( H  Fn  A  ->  H  =  ( x  e.  A  |->  ( H `  x ) ) )
31, 2syl 14 . . 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 3317 . . . 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 6042 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( B R C ) ) )
113, 10eqeq12d 2172 . 2  |-  ( ph  ->  ( H  =  ( F  oF R G )  <->  ( x  e.  A  |->  ( H `
 x ) )  =  ( x  e.  A  |->  ( B R C ) ) ) )
12 funfvex 5488 . . . . . 6  |-  ( ( Fun  H  /\  x  e.  dom  H )  -> 
( H `  x
)  e.  _V )
1312funfni 5273 . . . . 5  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( H `  x
)  e.  _V )
141, 13sylan 281 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( H `  x )  e.  _V )
1514ralrimiva 2530 . . 3  |-  ( ph  ->  A. x  e.  A  ( H `  x )  e.  _V )
16 mpteqb 5561 . . 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 ) ) )
1715, 16syl 14 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  ( H `  x ) )  =  ( x  e.  A  |->  ( B R C ) )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
1811, 17bitrd 187 1  |-  ( ph  ->  ( H  =  ( F  oF R G )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   A.wral 2435   _Vcvv 2712    |-> cmpt 4028    Fn wfn 5168   ` cfv 5173  (class class class)co 5827    oFcof 6033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-pow 4138  ax-pr 4172  ax-setind 4499
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-ov 5830  df-oprab 5831  df-mpo 5832  df-of 6035
This theorem is referenced by: (None)
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