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Theorem caofrss 6329
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  o R R G  ->  F  o R 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 5862 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
3 caofcom.3 . . . . 5  |-  ( ph  ->  G : A --> S )
43ffvelrnda 5862 . . . 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 2790 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T y ) )
76adantr 452 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  A. y  e.  S  ( x R y  ->  x T y ) )
8 breq1 4207 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x R y  <->  ( F `  w ) R y ) )
9 breq1 4207 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x T y  <->  ( F `  w ) T y ) )
108, 9imbi12d 312 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( x R y  ->  x T y )  <->  ( ( F `
 w ) R y  ->  ( F `  w ) T y ) ) )
11 breq2 4208 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y  <->  ( F `  w ) R ( G `  w ) ) )
12 breq2 4208 . . . . . 6  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) T y  <->  ( F `  w ) T ( G `  w ) ) )
1311, 12imbi12d 312 . . . . 5  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y  ->  ( F `  w ) T y )  <->  ( ( F `
 w ) R ( G `  w
)  ->  ( F `  w ) T ( G `  w ) ) ) )
1410, 13rspc2va 3051 . . . 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 1183 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w )  -> 
( F `  w
) T ( G `
 w ) ) )
1615ralimdva 2776 . 2  |-  ( ph  ->  ( A. w  e.  A  ( F `  w ) R ( G `  w )  ->  A. w  e.  A  ( F `  w ) T ( G `  w ) ) )
17 ffn 5583 . . . 4  |-  ( F : A --> S  ->  F  Fn  A )
181, 17syl 16 . . 3  |-  ( ph  ->  F  Fn  A )
19 ffn 5583 . . . 4  |-  ( G : A --> S  ->  G  Fn  A )
203, 19syl 16 . . 3  |-  ( ph  ->  G  Fn  A )
21 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
22 inidm 3542 . . 3  |-  ( A  i^i  A )  =  A
23 eqidd 2436 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
24 eqidd 2436 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( G `  w ) )
2518, 20, 21, 21, 22, 23, 24ofrfval 6305 . 2  |-  ( ph  ->  ( F  o R R G  <->  A. w  e.  A  ( F `  w ) R ( G `  w ) ) )
2618, 20, 21, 21, 22, 23, 24ofrfval 6305 . 2  |-  ( ph  ->  ( F  o R T G  <->  A. w  e.  A  ( F `  w ) T ( G `  w ) ) )
2716, 25, 263imtr4d 260 1  |-  ( ph  ->  ( F  o R R G  ->  F  o R T G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204    Fn wfn 5441   -->wf 5442   ` cfv 5446    o Rcofr 6296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ofr 6298
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