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Theorem caofref 6006
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 3945 . . . 4  |-  ( x  =  ( F `  w )  ->  (
x R x  <->  ( F `  w ) R ( F `  w ) ) )
3 caofref.3 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  x R x )
43ralrimiva 2505 . . . . 5  |-  ( ph  ->  A. x  e.  S  x R x )
54adantr 274 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  x R x )
6 caofref.2 . . . . 5  |-  ( ph  ->  F : A --> S )
76ffvelrnda 5558 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
82, 5, 7rspcdva 2794 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w ) R ( F `  w ) )
98ralrimiva 2505 . 2  |-  ( ph  ->  A. w  e.  A  ( F `  w ) R ( F `  w ) )
10 ffn 5275 . . . 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 3285 . . 3  |-  ( A  i^i  A )  =  A
14 eqidd 2140 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
1511, 11, 12, 12, 13, 14, 14ofrfval 5993 . 2  |-  ( ph  ->  ( F  oR R F  <->  A. w  e.  A  ( F `  w ) R ( F `  w ) ) )
169, 15mpbird 166 1  |-  ( ph  ->  F  oR R F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2416   class class class wbr 3932    Fn wfn 5121   -->wf 5122   ` cfv 5126    oRcofr 5984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-ofr 5986
This theorem is referenced by: (None)
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