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Theorem caofref 5876
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 id 19 . . . . 5  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
21, 1breq12d 3858 . . . 4  |-  ( x  =  ( F `  w )  ->  (
x R x  <->  ( F `  w ) R ( F `  w ) ) )
3 caofref.3 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  x R x )
43ralrimiva 2446 . . . . 5  |-  ( ph  ->  A. x  e.  S  x R x )
54adantr 270 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  x R x )
6 caofref.2 . . . . 5  |-  ( ph  ->  F : A --> S )
76ffvelrnda 5434 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
82, 5, 7rspcdva 2727 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w ) R ( F `  w ) )
98ralrimiva 2446 . 2  |-  ( ph  ->  A. w  e.  A  ( F `  w ) R ( F `  w ) )
10 ffn 5161 . . . 4  |-  ( F : A --> S  ->  F  Fn  A )
116, 10syl 14 . . 3  |-  ( ph  ->  F  Fn  A )
12 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
13 inidm 3209 . . 3  |-  ( A  i^i  A )  =  A
14 eqidd 2089 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
1511, 11, 12, 12, 13, 14, 14ofrfval 5864 . 2  |-  ( ph  ->  ( F  oR R F  <->  A. w  e.  A  ( F `  w ) R ( F `  w ) ) )
169, 15mpbird 165 1  |-  ( ph  ->  F  oR R F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   class class class wbr 3845    Fn wfn 5010   -->wf 5011   ` cfv 5015    oRcofr 5855
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ofr 5857
This theorem is referenced by: (None)
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