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Theorem caofcan 27518
Description: Transfer a cancellation law like mulcan 9660 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
Allowed substitution hints:    A( x, y, z)    V( x, y, z)

Proof of Theorem caofcan
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofcan.2 . . . . . . 7  |-  ( ph  ->  F : A --> T )
2 ffn 5592 . . . . . . 7  |-  ( F : A --> T  ->  F  Fn  A )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  F  Fn  A )
4 caofcan.3 . . . . . . 7  |-  ( ph  ->  G : A --> S )
5 ffn 5592 . . . . . . 7  |-  ( G : A --> S  ->  G  Fn  A )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  G  Fn  A )
7 caofcan.1 . . . . . 6  |-  ( ph  ->  A  e.  V )
8 inidm 3551 . . . . . 6  |-  ( A  i^i  A )  =  A
9 eqidd 2438 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
10 eqidd 2438 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
113, 6, 7, 7, 8, 9, 10ofval 6315 . . . . 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 5592 . . . . . . 7  |-  ( H : A --> S  ->  H  Fn  A )
1412, 13syl 16 . . . . . 6  |-  ( ph  ->  H  Fn  A )
15 eqidd 2438 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  =  ( H `  w ) )
163, 14, 7, 7, 8, 9, 15ofval 6315 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( F  o F R H ) `  w )  =  ( ( F `  w
) R ( H `
 w ) ) )
1711, 16eqeq12d 2451 . . . 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 )
191ffvelrnda 5871 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  T )
204ffvelrnda 5871 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
2112ffvelrnda 5871 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
22 caofcan.5 . . . . . 6  |-  ( (
ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x R y )  =  ( x R z )  <-> 
y  =  z ) )
2322caovcang 6249 . . . . 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 ) ) )
2418, 19, 20, 21, 23syl13anc 1187 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F `  w ) R ( G `  w ) )  =  ( ( F `  w ) R ( H `  w ) )  <->  ( G `  w )  =  ( H `  w ) ) )
2517, 24bitrd 246 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F  o F R G ) `  w )  =  ( ( F  o F R H ) `  w )  <->  ( G `  w )  =  ( H `  w ) ) )
2625ralbidva 2722 . 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 ) ) )
273, 6, 7, 7, 8offn 6317 . . 3  |-  ( ph  ->  ( F  o F R G )  Fn  A )
283, 14, 7, 7, 8offn 6317 . . 3  |-  ( ph  ->  ( F  o F R H )  Fn  A )
29 eqfnfv 5828 . . 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 ) ) )
3027, 28, 29syl2anc 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 ) ) )
31 eqfnfv 5828 . . 3  |-  ( ( G  Fn  A  /\  H  Fn  A )  ->  ( G  =  H  <->  A. w  e.  A  ( G `  w )  =  ( H `  w ) ) )
326, 14, 31syl2anc 644 . 2  |-  ( ph  ->  ( G  =  H  <->  A. w  e.  A  ( G `  w )  =  ( H `  w ) ) )
3326, 30, 323bitr4d 278 1  |-  ( ph  ->  ( ( F  o F R G )  =  ( F  o F R H )  <->  G  =  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082    o Fcof 6304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306
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