Users' Mathboxes Mathbox for Steve Rodriguez < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  caofcan Unicode version

Theorem caofcan 26939
Description: Transfer a cancellation law like mulcan 9400 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
Hypotheses
Ref Expression
caofcan.1  |-  ( ph  ->  A  e.  V )
caofcan.2  |-  ( ph  ->  F : A --> T )
caofcan.3  |-  ( ph  ->  G : A --> S )
caofcan.4  |-  ( ph  ->  H : A --> S )
caofcan.5  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x R y )  =  ( x R z )  <-> 
y  =  z ) )
Assertion
Ref Expression
caofcan  |-  ( ph  ->  ( ( F  o F R G )  =  ( F  o F R H )  <->  G  =  H ) )
Distinct variable groups:    x, y,
z, F    x, G, y, z    x, H, y, z    x, R, y, z    ph, x, y, z   
x, S, y, z   
x, T, y, z
Dummy variable  w is distinct from all other variables.
Allowed substitution hints:    A( x, y, z)    V( x, y, z)

Proof of Theorem caofcan
StepHypRef Expression
1 caofcan.2 . . . . . . 7  |-  ( ph  ->  F : A --> T )
2 ffn 5354 . . . . . . 7  |-  ( F : A --> T  ->  F  Fn  A )
31, 2syl 17 . . . . . 6  |-  ( ph  ->  F  Fn  A )
4 caofcan.3 . . . . . . 7  |-  ( ph  ->  G : A --> S )
5 ffn 5354 . . . . . . 7  |-  ( G : A --> S  ->  G  Fn  A )
64, 5syl 17 . . . . . 6  |-  ( ph  ->  G  Fn  A )
7 caofcan.1 . . . . . 6  |-  ( ph  ->  A  e.  V )
8 inidm 3379 . . . . . 6  |-  ( A  i^i  A )  =  A
9 eqidd 2285 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
10 eqidd 2285 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
113, 6, 7, 7, 8, 9, 10ofval 6048 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( F  o F R G ) `  w )  =  ( ( F `  w
) R ( G `
 w ) ) )
12 caofcan.4 . . . . . . 7  |-  ( ph  ->  H : A --> S )
13 ffn 5354 . . . . . . 7  |-  ( H : A --> S  ->  H  Fn  A )
1412, 13syl 17 . . . . . 6  |-  ( ph  ->  H  Fn  A )
15 eqidd 2285 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  =  ( H `  w ) )
163, 14, 7, 7, 8, 9, 15ofval 6048 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( F  o F R H ) `  w )  =  ( ( F `  w
) R ( H `
 w ) ) )
1711, 16eqeq12d 2298 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F  o F R G ) `  w )  =  ( ( F  o F R H ) `  w )  <->  ( ( F `  w ) R ( G `  w ) )  =  ( ( F `  w ) R ( H `  w ) ) ) )
18 simpl 445 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ph )
19 ffvelrn 5624 . . . . . 6  |-  ( ( F : A --> T  /\  w  e.  A )  ->  ( F `  w
)  e.  T )
201, 19sylan 459 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  T )
21 ffvelrn 5624 . . . . . 6  |-  ( ( G : A --> S  /\  w  e.  A )  ->  ( G `  w
)  e.  S )
224, 21sylan 459 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
23 ffvelrn 5624 . . . . . 6  |-  ( ( H : A --> S  /\  w  e.  A )  ->  ( H `  w
)  e.  S )
2412, 23sylan 459 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
25 caofcan.5 . . . . . 6  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x R y )  =  ( x R z )  <-> 
y  =  z ) )
2625caovcang 5982 . . . . 5  |-  ( (
ph  /\  ( ( F `  w )  e.  T  /\  ( G `  w )  e.  S  /\  ( H `  w )  e.  S ) )  -> 
( ( ( F `
 w ) R ( G `  w
) )  =  ( ( F `  w
) R ( H `
 w ) )  <-> 
( G `  w
)  =  ( H `
 w ) ) )
2718, 20, 22, 24, 26syl13anc 1186 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F `  w ) R ( G `  w ) )  =  ( ( F `  w ) R ( H `  w ) )  <->  ( G `  w )  =  ( H `  w ) ) )
2817, 27bitrd 246 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F  o F R G ) `  w )  =  ( ( F  o F R H ) `  w )  <->  ( G `  w )  =  ( H `  w ) ) )
2928ralbidva 2560 . 2  |-  ( ph  ->  ( A. w  e.  A  ( ( F  o F R G ) `  w )  =  ( ( F  o F R H ) `  w )  <->  A. w  e.  A  ( G `  w )  =  ( H `  w ) ) )
303, 6, 7, 7, 8offn 6050 . . 3  |-  ( ph  ->  ( F  o F R G )  Fn  A )
313, 14, 7, 7, 8offn 6050 . . 3  |-  ( ph  ->  ( F  o F R H )  Fn  A )
32 eqfnfv 5583 . . 3  |-  ( ( ( F  o F R G )  Fn  A  /\  ( F  o F R H )  Fn  A )  ->  ( ( F  o F R G )  =  ( F  o F R H )  <->  A. w  e.  A  ( ( F  o F R G ) `  w )  =  ( ( F  o F R H ) `  w ) ) )
3330, 31, 32syl2anc 644 . 2  |-  ( ph  ->  ( ( F  o F R G )  =  ( F  o F R H )  <->  A. w  e.  A  ( ( F  o F R G ) `  w )  =  ( ( F  o F R H ) `  w ) ) )
34 eqfnfv 5583 . . 3  |-  ( ( G  Fn  A  /\  H  Fn  A )  ->  ( G  =  H  <->  A. w  e.  A  ( G `  w )  =  ( H `  w ) ) )
356, 14, 34syl2anc 644 . 2  |-  ( ph  ->  ( G  =  H  <->  A. w  e.  A  ( G `  w )  =  ( H `  w ) ) )
3629, 33, 353bitr4d 278 1  |-  ( ph  ->  ( ( F  o F R G )  =  ( F  o F R H )  <->  G  =  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   A.wral 2544    Fn wfn 5216   -->wf 5217   ` cfv 5221  (class class class)co 5819    o Fcof 6037
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039
  Copyright terms: Public domain W3C validator