MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  offveqb Structured version   Unicode version

Theorem offveqb 6498
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 dffn5 5848 . . . 4  |-  ( H  Fn  A  <->  H  =  ( x  e.  A  |->  ( H `  x
) ) )
31, 2sylib 196 . . 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 3645 . . . 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 6482 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( B R C ) ) )
113, 10eqeq12d 2422 . 2  |-  ( ph  ->  ( H  =  ( F  oF R G )  <->  ( x  e.  A  |->  ( H `
 x ) )  =  ( x  e.  A  |->  ( B R C ) ) ) )
12 fvex 5813 . . . . 5  |-  ( H `
 x )  e. 
_V
1312a1i 11 . . . 4  |-  ( ph  ->  ( H `  x
)  e.  _V )
1413ralrimivw 2816 . . 3  |-  ( ph  ->  A. x  e.  A  ( H `  x )  e.  _V )
15 mpteqb 5902 . . 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 17 . 2  |-  ( ph  ->  ( ( x  e.  A  |->  ( H `  x ) )  =  ( x  e.  A  |->  ( B R C ) )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
1711, 16bitrd 253 1  |-  ( ph  ->  ( H  =  ( F  oF R G )  <->  A. x  e.  A  ( H `  x )  =  ( B R C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840   A.wral 2751   _Vcvv 3056    |-> cmpt 4450    Fn wfn 5518   ` cfv 5523  (class class class)co 6232    oFcof 6473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475
This theorem is referenced by:  eqlkr2  32082
  Copyright terms: Public domain W3C validator