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Theorem dffrege115 36426
Description: If from the the circumstance that  c is a result of an application of the procedure  R to  b, whatever  b may be, it can be inferred that every result of an application of the procedure  R to  b is the same as  c, then we say : "The procedure 
R is single-valued". Definition 115 of [Frege1879] p. 77. (Contributed by RP, 7-Jul-2020.)
Assertion
Ref Expression
dffrege115  |-  ( A. c A. b ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  Fun  `' `' R )
Distinct variable group:    a, b, c, R

Proof of Theorem dffrege115
StepHypRef Expression
1 alcom 1894 . 2  |-  ( A. c A. b ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  A. b A. c ( b R c  ->  A. a
( b R a  ->  a  =  c ) ) )
2 19.21v 1775 . . . . . . 7  |-  ( A. a ( b R c  ->  ( b R a  ->  a  =  c ) )  <-> 
( b R c  ->  A. a ( b R a  ->  a  =  c ) ) )
3 impexp 447 . . . . . . . . 9  |-  ( ( ( b R c  /\  b R a )  ->  a  =  c )  <->  ( b R c  ->  (
b R a  -> 
a  =  c ) ) )
4 vex 3081 . . . . . . . . . . . . 13  |-  b  e. 
_V
5 vex 3081 . . . . . . . . . . . . 13  |-  c  e. 
_V
64, 5brcnv 5029 . . . . . . . . . . . 12  |-  ( b `' `' R c  <->  c `' R b )
7 df-br 4418 . . . . . . . . . . . 12  |-  ( b `' `' R c  <->  <. b ,  c >.  e.  `' `' R )
85, 4brcnv 5029 . . . . . . . . . . . 12  |-  ( c `' R b  <->  b R
c )
96, 7, 83bitr3ri 279 . . . . . . . . . . 11  |-  ( b R c  <->  <. b ,  c >.  e.  `' `' R )
10 vex 3081 . . . . . . . . . . . . 13  |-  a  e. 
_V
114, 10brcnv 5029 . . . . . . . . . . . 12  |-  ( b `' `' R a  <->  a `' R b )
12 df-br 4418 . . . . . . . . . . . 12  |-  ( b `' `' R a  <->  <. b ,  a >.  e.  `' `' R )
1310, 4brcnv 5029 . . . . . . . . . . . 12  |-  ( a `' R b  <->  b R
a )
1411, 12, 133bitr3ri 279 . . . . . . . . . . 11  |-  ( b R a  <->  <. b ,  a >.  e.  `' `' R )
159, 14anbi12ci 702 . . . . . . . . . 10  |-  ( ( b R c  /\  b R a )  <->  ( <. b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R ) )
1615imbi1i 326 . . . . . . . . 9  |-  ( ( ( b R c  /\  b R a )  ->  a  =  c )  <->  ( ( <. b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c ) )
173, 16bitr3i 254 . . . . . . . 8  |-  ( ( b R c  -> 
( b R a  ->  a  =  c ) )  <->  ( ( <. b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c ) )
1817albii 1687 . . . . . . 7  |-  ( A. a ( b R c  ->  ( b R a  ->  a  =  c ) )  <->  A. a ( ( <.
b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c ) )
192, 18bitr3i 254 . . . . . 6  |-  ( ( b R c  ->  A. a ( b R a  ->  a  =  c ) )  <->  A. a
( ( <. b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c ) )
2019albii 1687 . . . . 5  |-  ( A. c ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  A. c A. a ( ( <.
b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c ) )
21 alcom 1894 . . . . 5  |-  ( A. c A. a ( (
<. b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c )  <->  A. a A. c ( ( <.
b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c ) )
2220, 21bitri 252 . . . 4  |-  ( A. c ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  A. a A. c ( ( <.
b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c ) )
23 opeq2 4182 . . . . . 6  |-  ( a  =  c  ->  <. b ,  a >.  =  <. b ,  c >. )
2423eleq1d 2489 . . . . 5  |-  ( a  =  c  ->  ( <. b ,  a >.  e.  `' `' R  <->  <. b ,  c
>.  e.  `' `' R
) )
2524mo4 2311 . . . 4  |-  ( E* a <. b ,  a
>.  e.  `' `' R  <->  A. a A. c ( ( <. b ,  a
>.  e.  `' `' R  /\  <. b ,  c
>.  e.  `' `' R
)  ->  a  =  c ) )
26 mo2v 2270 . . . 4  |-  ( E* a <. b ,  a
>.  e.  `' `' R  <->  E. c A. a (
<. b ,  a >.  e.  `' `' R  ->  a  =  c ) )
2722, 25, 263bitr2i 276 . . 3  |-  ( A. c ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  E. c A. a ( <. b ,  a >.  e.  `' `' R  ->  a  =  c ) )
2827albii 1687 . 2  |-  ( A. b A. c ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  A. b E. c A. a (
<. b ,  a >.  e.  `' `' R  ->  a  =  c ) )
29 relcnv 5219 . . . 4  |-  Rel  `' `' R
3029biantrur 508 . . 3  |-  ( A. b E. c A. a
( <. b ,  a
>.  e.  `' `' R  ->  a  =  c )  <-> 
( Rel  `' `' R  /\  A. b E. c A. a (
<. b ,  a >.  e.  `' `' R  ->  a  =  c ) ) )
31 dffun5 5606 . . 3  |-  ( Fun  `' `' R  <->  ( Rel  `' `' R  /\  A. b E. c A. a (
<. b ,  a >.  e.  `' `' R  ->  a  =  c ) ) )
3230, 31bitr4i 255 . 2  |-  ( A. b E. c A. a
( <. b ,  a
>.  e.  `' `' R  ->  a  =  c )  <->  Fun  `' `' R )
331, 28, 323bitri 274 1  |-  ( A. c A. b ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  Fun  `' `' R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1659    e. wcel 1867   E*wmo 2264   <.cop 3999   class class class wbr 4417   `'ccnv 4845   Rel wrel 4851   Fun wfun 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-fun 5595
This theorem is referenced by:  frege116  36427
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