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Theorem offveqb 5866
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 5344 . . . 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 3209 . . . 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 5855 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( B R C ) ) )
113, 10eqeq12d 2102 . 2  |-  ( ph  ->  ( H  =  ( F  oF R G )  <->  ( x  e.  A  |->  ( H `
 x ) )  =  ( x  e.  A  |->  ( B R C ) ) ) )
12 funfvex 5316 . . . . . 6  |-  ( ( Fun  H  /\  x  e.  dom  H )  -> 
( H `  x
)  e.  _V )
1312funfni 5108 . . . . 5  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( H `  x
)  e.  _V )
141, 13sylan 277 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( H `  x )  e.  _V )
1514ralrimiva 2446 . . 3  |-  ( ph  ->  A. x  e.  A  ( H `  x )  e.  _V )
16 mpteqb 5387 . . 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 186 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   _Vcvv 2619    |-> cmpt 3897    Fn wfn 5005   ` cfv 5010  (class class class)co 5644    oFcof 5846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-setind 4351
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-of 5848
This theorem is referenced by: (None)
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