ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  offeq Unicode version

Theorem offeq 5988
Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
Hypotheses
Ref Expression
off.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  T ) )  -> 
( x R y )  e.  U )
off.2  |-  ( ph  ->  F : A --> S )
off.3  |-  ( ph  ->  G : B --> T )
off.4  |-  ( ph  ->  A  e.  V )
off.5  |-  ( ph  ->  B  e.  W )
off.6  |-  ( A  i^i  B )  =  C
offeq.4  |-  ( ph  ->  H : C --> U )
offeq.5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  D )
offeq.6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  E )
offeq.7  |-  ( (
ph  /\  x  e.  C )  ->  ( D R E )  =  ( H `  x
) )
Assertion
Ref Expression
offeq  |-  ( ph  ->  ( F  oF R G )  =  H )
Distinct variable groups:    y, G, x    ph, x, y    x, S, y    x, T, y   
x, F, y    x, R, y    x, U, y   
x, H    x, G    x, C
Allowed substitution hints:    A( x, y)    B( x, y)    C( y)    D( x, y)    E( x, y)    H( y)    V( x, y)    W( x, y)

Proof of Theorem offeq
StepHypRef Expression
1 off.1 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  T ) )  -> 
( x R y )  e.  U )
2 off.2 . . . 4  |-  ( ph  ->  F : A --> S )
3 off.3 . . . 4  |-  ( ph  ->  G : B --> T )
4 off.4 . . . 4  |-  ( ph  ->  A  e.  V )
5 off.5 . . . 4  |-  ( ph  ->  B  e.  W )
6 off.6 . . . 4  |-  ( A  i^i  B )  =  C
71, 2, 3, 4, 5, 6off 5987 . . 3  |-  ( ph  ->  ( F  oF R G ) : C --> U )
87ffnd 5268 . 2  |-  ( ph  ->  ( F  oF R G )  Fn  C )
9 offeq.4 . . 3  |-  ( ph  ->  H : C --> U )
109ffnd 5268 . 2  |-  ( ph  ->  H  Fn  C )
112ffnd 5268 . . . 4  |-  ( ph  ->  F  Fn  A )
123ffnd 5268 . . . 4  |-  ( ph  ->  G  Fn  B )
13 offeq.5 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  D )
14 offeq.6 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  E )
15 offeq.7 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( D R E )  =  ( H `  x
) )
169ffvelrnda 5548 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( H `  x )  e.  U )
1715, 16eqeltrd 2214 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  ( D R E )  e.  U )
1811, 12, 4, 5, 6, 13, 14, 17ofvalg 5984 . . 3  |-  ( (
ph  /\  x  e.  C )  ->  (
( F  oF R G ) `  x )  =  ( D R E ) )
1918, 15eqtrd 2170 . 2  |-  ( (
ph  /\  x  e.  C )  ->  (
( F  oF R G ) `  x )  =  ( H `  x ) )
208, 10, 19eqfnfvd 5514 1  |-  ( ph  ->  ( F  oF R G )  =  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    i^i cin 3065   -->wf 5114   ` cfv 5118  (class class class)co 5767    oFcof 5973
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-of 5975
This theorem is referenced by:  dviaddf  12827
  Copyright terms: Public domain W3C validator