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Theorem caofref 6103
Description: Transfer a reflexive 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 )
caofref.3  |-  ( (
ph  /\  x  e.  S )  ->  x R x )
Assertion
Ref Expression
caofref  |-  ( ph  ->  F  o R R F )
Distinct variable groups:    x, F    ph, x    x, R    x, S
Allowed substitution hints:    A( x)    V( x)

Proof of Theorem caofref
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . 5  |-  ( ph  ->  F : A --> S )
2 ffvelrn 5663 . . . . 5  |-  ( ( F : A --> S  /\  w  e.  A )  ->  ( F `  w
)  e.  S )
31, 2sylan 457 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
4 caofref.3 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  x R x )
54ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. x  e.  S  x R x )
65adantr 451 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  x R x )
7 id 19 . . . . . 6  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
87, 7breq12d 4036 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
x R x  <->  ( F `  w ) R ( F `  w ) ) )
98rspcv 2880 . . . 4  |-  ( ( F `  w )  e.  S  ->  ( A. x  e.  S  x R x  ->  ( F `  w ) R ( F `  w ) ) )
103, 6, 9sylc 56 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w ) R ( F `  w ) )
1110ralrimiva 2626 . 2  |-  ( ph  ->  A. w  e.  A  ( F `  w ) R ( F `  w ) )
12 ffn 5389 . . . 4  |-  ( F : A --> S  ->  F  Fn  A )
131, 12syl 15 . . 3  |-  ( ph  ->  F  Fn  A )
14 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
15 inidm 3378 . . 3  |-  ( A  i^i  A )  =  A
16 eqidd 2284 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
1713, 13, 14, 14, 15, 16, 16ofrfval 6086 . 2  |-  ( ph  ->  ( F  o R R F  <->  A. w  e.  A  ( F `  w ) R ( F `  w ) ) )
1811, 17mpbird 223 1  |-  ( ph  ->  F  o R R F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    Fn wfn 5250   -->wf 5251   ` cfv 5255    o Rcofr 6077
This theorem is referenced by:  psrridm  16149  itg2itg1  19091  itg20  19092
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-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-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-ofr 6079
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