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Theorem caofcan 26706
Description: Transfer a cancellation law like mulcan 9285 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
StepHypRef Expression
1 caofcan.2 . . . . . . 7  |-  ( ph  ->  F : A --> T )
2 ffn 5246 . . . . . . 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 5246 . . . . . . 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 3285 . . . . . 6  |-  ( A  i^i  A )  =  A
9 eqidd 2254 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
10 eqidd 2254 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
113, 6, 7, 7, 8, 9, 10ofval 5939 . . . . 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 5246 . . . . . . 7  |-  ( H : A --> S  ->  H  Fn  A )
1412, 13syl 17 . . . . . 6  |-  ( ph  ->  H  Fn  A )
15 eqidd 2254 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  =  ( H `  w ) )
163, 14, 7, 7, 8, 9, 15ofval 5939 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( F  o F R H ) `  w )  =  ( ( F `  w
) R ( H `
 w ) ) )
1711, 16eqeq12d 2267 . . . 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 5515 . . . . . 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 5515 . . . . . 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 5515 . . . . . 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 5873 . . . . 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 1189 . . . 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 2523 . 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 5941 . . 3  |-  ( ph  ->  ( F  o F R G )  Fn  A )
313, 14, 7, 7, 8offn 5941 . . 3  |-  ( ph  ->  ( F  o F R H )  Fn  A )
32 eqfnfv 5474 . . 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 645 . 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 5474 . . 3  |-  ( ( G  Fn  A  /\  H  Fn  A )  ->  ( G  =  H  <->  A. w  e.  A  ( G `  w )  =  ( H `  w ) ) )
356, 14, 34syl2anc 645 . 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 939    = wceq 1619    e. wcel 1621   A.wral 2509    Fn wfn 4587   -->wf 4588   ` cfv 4592  (class class class)co 5710    o Fcof 5928
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930
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