Theorem dffrege115 38272

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

Step	Hyp	Ref	Expression
1		alcom 2037	. 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 1868	. . . . . . 7 |- ( A. a ( b R c -> ( b R a -> a = c ) ) <-> ( b R c -> A. a ( b R a -> a = c ) ) )
3		impexp 462	. . . . . . . . 9 |- ( ( ( b R c /\ b R a ) -> a = c ) <-> ( b R c -> ( b R a -> a = c ) ) )
4		vex 3203	. . . . . . . . . . . . 13 |- b e. _V
5		vex 3203	. . . . . . . . . . . . 13 |- c e. _V
6	4, 5	brcnv 5305	. . . . . . . . . . . 12 |- ( b `' `' R c <-> c `' R b )
7		df-br 4654	. . . . . . . . . . . 12 |- ( b `' `' R c <-> <. b , c >. e. `' `' R )
8	5, 4	brcnv 5305	. . . . . . . . . . . 12 |- ( c `' R b <-> b R c )
9	6, 7, 8	3bitr3ri 291	. . . . . . . . . . 11 |- ( b R c <-> <. b , c >. e. `' `' R )
10		vex 3203	. . . . . . . . . . . . 13 |- a e. _V
11	4, 10	brcnv 5305	. . . . . . . . . . . 12 |- ( b `' `' R a <-> a `' R b )
12		df-br 4654	. . . . . . . . . . . 12 |- ( b `' `' R a <-> <. b , a >. e. `' `' R )
13	10, 4	brcnv 5305	. . . . . . . . . . . 12 |- ( a `' R b <-> b R a )
14	11, 12, 13	3bitr3ri 291	. . . . . . . . . . 11 |- ( b R a <-> <. b , a >. e. `' `' R )
15	9, 14	anbi12ci 734	. . . . . . . . . 10 |- ( ( b R c /\ b R a ) <-> ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) )
16	15	imbi1i 339	. . . . . . . . 9 |- ( ( ( b R c /\ b R a ) -> a = c ) <-> ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
17	3, 16	bitr3i 266	. . . . . . . 8 |- ( ( b R c -> ( b R a -> a = c ) ) <-> ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
18	17	albii 1747	. . . . . . 7 |- ( A. a ( b R c -> ( b R a -> a = c ) ) <-> A. a ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
19	2, 18	bitr3i 266	. . . . . 6 |- ( ( b R c -> A. a ( b R a -> a = c ) ) <-> A. a ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
20	19	albii 1747	. . . . 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 2037	. . . . 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 ) )
22	20, 21	bitri 264	. . . 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 4403	. . . . . 6 |- ( a = c -> <. b , a >. = <. b , c >. )
24	23	eleq1d 2686	. . . . 5 |- ( a = c -> ( <. b , a >. e. `' `' R <-> <. b , c >. e. `' `' R ) )
25	24	mo4 2517	. . . 4 |- ( E* a <. b , a >. e. `' `' R <-> A. a A. c ( ( <. b , a >. e. `' `' R /\ <. b , c >. e. `' `' R ) -> a = c ) )
26		mo2v 2477	. . . 4 |- ( E* a <. b , a >. e. `' `' R <-> E. c A. a ( <. b , a >. e. `' `' R -> a = c ) )
27	22, 25, 26	3bitr2i 288	. . 3 |- ( A. c ( b R c -> A. a ( b R a -> a = c ) ) <-> E. c A. a ( <. b , a >. e. `' `' R -> a = c ) )
28	27	albii 1747	. 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 5503	. . . 4 |- Rel `' `' R
30	29	biantrur 527	. . 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 5901	. . 3 |- ( Fun `' `' R <-> ( Rel `' `' R /\ A. b E. c A. a ( <. b , a >. e. `' `' R -> a = c ) ) )
32	30, 31	bitr4i 267	. 2 |- ( A. b E. c A. a ( <. b , a >. e. `' `' R -> a = c ) <-> Fun `' `' R )
33	1, 28, 32	3bitri 286	1 |- ( A. c A. b ( b R c -> A. a ( b R a -> a = c ) ) <-> Fun `' `' R )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   E*wmo 2471   <.cop 4183   class class class wbr 4653   `'ccnv 5113   Rel wrel 5119   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  frege116  38273