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Theorem caofrss 5974
Description: Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1  |-  ( ph  ->  A  e.  V )
caofref.2  |-  ( ph  ->  F : A --> S )
caofcom.3  |-  ( ph  ->  G : A --> S )
caofrss.4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x R y  ->  x T y ) )
Assertion
Ref Expression
caofrss  |-  ( ph  ->  ( F  oR R G  ->  F  oR T G ) )
Distinct variable groups:    x, y, F   
x, G, y    ph, x, y    x, R, y    x, S, y    x, T, y
Allowed substitution hints:    A( x, y)    V( x, y)

Proof of Theorem caofrss
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . 5  |-  ( ph  ->  F : A --> S )
21ffvelrnda 5523 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
3 caofcom.3 . . . . 5  |-  ( ph  ->  G : A --> S )
43ffvelrnda 5523 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
5 caofrss.4 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x R y  ->  x T y ) )
65ralrimivva 2491 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T y ) )
76adantr 274 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T y ) )
8 breq1 3902 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x R y  <->  ( F `  w ) R y ) )
9 breq1 3902 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x T y  <->  ( F `  w ) T y ) )
108, 9imbi12d 233 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( x R y  ->  x T y )  <->  ( ( F `
 w ) R y  ->  ( F `  w ) T y ) ) )
11 breq2 3903 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y  <->  ( F `  w ) R ( G `  w ) ) )
12 breq2 3903 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) T y  <->  ( F `  w ) T ( G `  w ) ) )
1311, 12imbi12d 233 . . . . 5  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y  ->  ( F `  w ) T y )  <->  ( ( F `
 w ) R ( G `  w
)  ->  ( F `  w ) T ( G `  w ) ) ) )
1410, 13rspc2va 2777 . . . 4  |-  ( ( ( ( F `  w )  e.  S  /\  ( G `  w
)  e.  S )  /\  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T
y ) )  -> 
( ( F `  w ) R ( G `  w )  ->  ( F `  w ) T ( G `  w ) ) )
152, 4, 7, 14syl21anc 1200 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w )  -> 
( F `  w
) T ( G `
 w ) ) )
1615ralimdva 2476 . 2  |-  ( ph  ->  ( A. w  e.  A  ( F `  w ) R ( G `  w )  ->  A. w  e.  A  ( F `  w ) T ( G `  w ) ) )
17 ffn 5242 . . . 4  |-  ( F : A --> S  ->  F  Fn  A )
181, 17syl 14 . . 3  |-  ( ph  ->  F  Fn  A )
19 ffn 5242 . . . 4  |-  ( G : A --> S  ->  G  Fn  A )
203, 19syl 14 . . 3  |-  ( ph  ->  G  Fn  A )
21 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
22 inidm 3255 . . 3  |-  ( A  i^i  A )  =  A
23 eqidd 2118 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
24 eqidd 2118 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
2518, 20, 21, 21, 22, 23, 24ofrfval 5958 . 2  |-  ( ph  ->  ( F  oR R G  <->  A. w  e.  A  ( F `  w ) R ( G `  w ) ) )
2618, 20, 21, 21, 22, 23, 24ofrfval 5958 . 2  |-  ( ph  ->  ( F  oR T G  <->  A. w  e.  A  ( F `  w ) T ( G `  w ) ) )
2716, 25, 263imtr4d 202 1  |-  ( ph  ->  ( F  oR R G  ->  F  oR T G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   A.wral 2393   class class class wbr 3899    Fn wfn 5088   -->wf 5089   ` cfv 5093    oRcofr 5949
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ofr 5951
This theorem is referenced by: (None)
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