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Theorem caofref 6079
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 4000 . . . 4  |-  ( x  =  ( F `  w )  ->  (
x R x  <->  ( F `  w ) R ( F `  w ) ) )
3 caofref.3 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  x R x )
43ralrimiva 2543 . . . . 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 5628 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
82, 5, 7rspcdva 2839 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w ) R ( F `  w ) )
98ralrimiva 2543 . 2  |-  ( ph  ->  A. w  e.  A  ( F `  w ) R ( F `  w ) )
10 ffn 5345 . . . 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 3336 . . 3  |-  ( A  i^i  A )  =  A
14 eqidd 2171 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
1511, 11, 12, 12, 13, 14, 14ofrfval 6066 . 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 1348    e. wcel 2141   A.wral 2448   class class class wbr 3987    Fn wfn 5191   -->wf 5192   ` cfv 5196    oRcofr 6057
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ofr 6059
This theorem is referenced by: (None)
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