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Theorem caofcan 27540
Description: Transfer a cancellation law like mulcan 9405 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 5389 . . . . . . 7  |-  ( F : A --> T  ->  F  Fn  A )
31, 2syl 15 . . . . . 6  |-  ( ph  ->  F  Fn  A )
4 caofcan.3 . . . . . . 7  |-  ( ph  ->  G : A --> S )
5 ffn 5389 . . . . . . 7  |-  ( G : A --> S  ->  G  Fn  A )
64, 5syl 15 . . . . . 6  |-  ( ph  ->  G  Fn  A )
7 caofcan.1 . . . . . 6  |-  ( ph  ->  A  e.  V )
8 inidm 3378 . . . . . 6  |-  ( A  i^i  A )  =  A
9 eqidd 2284 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
10 eqidd 2284 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
113, 6, 7, 7, 8, 9, 10ofval 6087 . . . . 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 5389 . . . . . . 7  |-  ( H : A --> S  ->  H  Fn  A )
1412, 13syl 15 . . . . . 6  |-  ( ph  ->  H  Fn  A )
15 eqidd 2284 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  =  ( H `  w ) )
163, 14, 7, 7, 8, 9, 15ofval 6087 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( F  o F R H ) `  w )  =  ( ( F `  w
) R ( H `
 w ) ) )
1711, 16eqeq12d 2297 . . . 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 443 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ph )
19 ffvelrn 5663 . . . . . 6  |-  ( ( F : A --> T  /\  w  e.  A )  ->  ( F `  w
)  e.  T )
201, 19sylan 457 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  T )
21 ffvelrn 5663 . . . . . 6  |-  ( ( G : A --> S  /\  w  e.  A )  ->  ( G `  w
)  e.  S )
224, 21sylan 457 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
23 ffvelrn 5663 . . . . . 6  |-  ( ( H : A --> S  /\  w  e.  A )  ->  ( H `  w
)  e.  S )
2412, 23sylan 457 . . . . 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 6021 . . . . 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 1184 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F `  w ) R ( G `  w ) )  =  ( ( F `  w ) R ( H `  w ) )  <->  ( G `  w )  =  ( H `  w ) ) )
2817, 27bitrd 244 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F  o F R G ) `  w )  =  ( ( F  o F R H ) `  w )  <->  ( G `  w )  =  ( H `  w ) ) )
2928ralbidva 2559 . 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 6089 . . 3  |-  ( ph  ->  ( F  o F R G )  Fn  A )
313, 14, 7, 7, 8offn 6089 . . 3  |-  ( ph  ->  ( F  o F R H )  Fn  A )
32 eqfnfv 5622 . . 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 642 . 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 5622 . . 3  |-  ( ( G  Fn  A  /\  H  Fn  A )  ->  ( G  =  H  <->  A. w  e.  A  ( G `  w )  =  ( H `  w ) ) )
356, 14, 34syl2anc 642 . 2  |-  ( ph  ->  ( G  =  H  <->  A. w  e.  A  ( G `  w )  =  ( H `  w ) ) )
3629, 33, 353bitr4d 276 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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