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

Theorem offveqb 6539
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 5906 . . . 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 3702 . . . 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 6524 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( B R C ) ) )
113, 10eqeq12d 2484 . 2  |-  ( ph  ->  ( H  =  ( F  oF R G )  <->  ( x  e.  A  |->  ( H `
 x ) )  =  ( x  e.  A  |->  ( B R C ) ) ) )
12 fvex 5869 . . . . 5  |-  ( H `
 x )  e. 
_V
1312a1i 11 . . . 4  |-  ( ph  ->  ( H `  x
)  e.  _V )
1413ralrimivw 2874 . . 3  |-  ( ph  ->  A. x  e.  A  ( H `  x )  e.  _V )
15 mpteqb 5957 . . 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 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 369    = wceq 1374    e. wcel 1762   A.wral 2809   _Vcvv 3108    |-> cmpt 4500    Fn wfn 5576   ` cfv 5581  (class class class)co 6277    oFcof 6515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517
This theorem is referenced by:  eqlkr2  33774
  Copyright terms: Public domain W3C validator