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Theorem dffrege115 36574
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 1923 . 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 1786 . . . . . . 7  |-  ( A. a ( b R c  ->  ( b R a  ->  a  =  c ) )  <-> 
( b R c  ->  A. a ( b R a  ->  a  =  c ) ) )
3 impexp 448 . . . . . . . . 9  |-  ( ( ( b R c  /\  b R a )  ->  a  =  c )  <->  ( b R c  ->  (
b R a  -> 
a  =  c ) ) )
4 vex 3048 . . . . . . . . . . . . 13  |-  b  e. 
_V
5 vex 3048 . . . . . . . . . . . . 13  |-  c  e. 
_V
64, 5brcnv 5017 . . . . . . . . . . . 12  |-  ( b `' `' R c  <->  c `' R b )
7 df-br 4403 . . . . . . . . . . . 12  |-  ( b `' `' R c  <->  <. b ,  c >.  e.  `' `' R )
85, 4brcnv 5017 . . . . . . . . . . . 12  |-  ( c `' R b  <->  b R
c )
96, 7, 83bitr3ri 280 . . . . . . . . . . 11  |-  ( b R c  <->  <. b ,  c >.  e.  `' `' R )
10 vex 3048 . . . . . . . . . . . . 13  |-  a  e. 
_V
114, 10brcnv 5017 . . . . . . . . . . . 12  |-  ( b `' `' R a  <->  a `' R b )
12 df-br 4403 . . . . . . . . . . . 12  |-  ( b `' `' R a  <->  <. b ,  a >.  e.  `' `' R )
1310, 4brcnv 5017 . . . . . . . . . . . 12  |-  ( a `' R b  <->  b R
a )
1411, 12, 133bitr3ri 280 . . . . . . . . . . 11  |-  ( b R a  <->  <. b ,  a >.  e.  `' `' R )
159, 14anbi12ci 704 . . . . . . . . . 10  |-  ( ( b R c  /\  b R a )  <->  ( <. b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R ) )
1615imbi1i 327 . . . . . . . . 9  |-  ( ( ( b R c  /\  b R a )  ->  a  =  c )  <->  ( ( <. b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c ) )
173, 16bitr3i 255 . . . . . . . 8  |-  ( ( b R c  -> 
( b R a  ->  a  =  c ) )  <->  ( ( <. b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c ) )
1817albii 1691 . . . . . . 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 255 . . . . . 6  |-  ( ( b R c  ->  A. a ( b R a  ->  a  =  c ) )  <->  A. a
( ( <. b ,  a >.  e.  `' `' R  /\  <. b ,  c >.  e.  `' `' R )  ->  a  =  c ) )
2019albii 1691 . . . . 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 1923 . . . . 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 253 . . . 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 4167 . . . . . 6  |-  ( a  =  c  ->  <. b ,  a >.  =  <. b ,  c >. )
2423eleq1d 2513 . . . . 5  |-  ( a  =  c  ->  ( <. b ,  a >.  e.  `' `' R  <->  <. b ,  c
>.  e.  `' `' R
) )
2524mo4 2346 . . . 4  |-  ( E* a <. b ,  a
>.  e.  `' `' R  <->  A. a A. c ( ( <. b ,  a
>.  e.  `' `' R  /\  <. b ,  c
>.  e.  `' `' R
)  ->  a  =  c ) )
26 mo2v 2306 . . . 4  |-  ( E* a <. b ,  a
>.  e.  `' `' R  <->  E. c A. a (
<. b ,  a >.  e.  `' `' R  ->  a  =  c ) )
2722, 25, 263bitr2i 277 . . 3  |-  ( A. c ( b R c  ->  A. a
( b R a  ->  a  =  c ) )  <->  E. c A. a ( <. b ,  a >.  e.  `' `' R  ->  a  =  c ) )
2827albii 1691 . 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 5207 . . . 4  |-  Rel  `' `' R
3029biantrur 509 . . 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 5595 . . 3  |-  ( Fun  `' `' R  <->  ( Rel  `' `' R  /\  A. b E. c A. a (
<. b ,  a >.  e.  `' `' R  ->  a  =  c ) ) )
3230, 31bitr4i 256 . 2  |-  ( A. b E. c A. a
( <. b ,  a
>.  e.  `' `' R  ->  a  =  c )  <->  Fun  `' `' R )
331, 28, 323bitri 275 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 188    /\ wa 371   A.wal 1442   E.wex 1663    e. wcel 1887   E*wmo 2300   <.cop 3974   class class class wbr 4402   `'ccnv 4833   Rel wrel 4839   Fun wfun 5576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-fun 5584
This theorem is referenced by:  frege116  36575
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