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Theorem caofref 6330
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 )
21ffvelrnda 5870 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
3 caofref.3 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  x R x )
43ralrimiva 2789 . . . . 5  |-  ( ph  ->  A. x  e.  S  x R x )
54adantr 452 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  x R x )
6 id 20 . . . . . 6  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
76, 6breq12d 4225 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
x R x  <->  ( F `  w ) R ( F `  w ) ) )
87rspcv 3048 . . . 4  |-  ( ( F `  w )  e.  S  ->  ( A. x  e.  S  x R x  ->  ( F `  w ) R ( F `  w ) ) )
92, 5, 8sylc 58 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w ) R ( F `  w ) )
109ralrimiva 2789 . 2  |-  ( ph  ->  A. w  e.  A  ( F `  w ) R ( F `  w ) )
11 ffn 5591 . . . 4  |-  ( F : A --> S  ->  F  Fn  A )
121, 11syl 16 . . 3  |-  ( ph  ->  F  Fn  A )
13 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
14 inidm 3550 . . 3  |-  ( A  i^i  A )  =  A
15 eqidd 2437 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
1612, 12, 13, 13, 14, 15, 15ofrfval 6313 . 2  |-  ( ph  ->  ( F  o R R F  <->  A. w  e.  A  ( F `  w ) R ( F `  w ) ) )
1710, 16mpbird 224 1  |-  ( ph  ->  F  o R R F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   class class class wbr 4212    Fn wfn 5449   -->wf 5450   ` cfv 5454    o Rcofr 6304
This theorem is referenced by:  psrridm  16468  itg2itg1  19628  itg20  19629
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ofr 6306
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