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Theorem caofref 5757
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  oR 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 5328 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
3 caofref.3 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  x R x )
43ralrimiva 2435 . . . . 5  |-  ( ph  ->  A. x  e.  S  x R x )
54adantr 270 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  x R x )
6 id 19 . . . . . 6  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
76, 6breq12d 3800 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
x R x  <->  ( F `  w ) R ( F `  w ) ) )
87rspcv 2698 . . . 4  |-  ( ( F `  w )  e.  S  ->  ( A. x  e.  S  x R x  ->  ( F `  w ) R ( F `  w ) ) )
92, 5, 8sylc 61 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w ) R ( F `  w ) )
109ralrimiva 2435 . 2  |-  ( ph  ->  A. w  e.  A  ( F `  w ) R ( F `  w ) )
11 ffn 5071 . . . 4  |-  ( F : A --> S  ->  F  Fn  A )
121, 11syl 14 . . 3  |-  ( ph  ->  F  Fn  A )
13 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
14 inidm 3176 . . 3  |-  ( A  i^i  A )  =  A
15 eqidd 2083 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
1612, 12, 13, 13, 14, 15, 15ofrfval 5745 . 2  |-  ( ph  ->  ( F  oR R F  <->  A. w  e.  A  ( F `  w ) R ( F `  w ) ) )
1710, 16mpbird 165 1  |-  ( ph  ->  F  oR R F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2349   class class class wbr 3787    Fn wfn 4921   -->wf 4922   ` cfv 4926    oRcofr 5736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ofr 5738
This theorem is referenced by: (None)
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