Theorem frege96d 38041

Description: If C follows A in the transitive closure of R and B follows C in R, then B follows A in the transitive closure of R. Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 38253. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege96d.r	|- ( ph -> R e. _V )
frege96d.a	|- ( ph -> A e. _V )
frege96d.b	|- ( ph -> B e. _V )
frege96d.c	|- ( ph -> C e. _V )
frege96d.ac	|- ( ph -> A ( t+ ` R ) C )
frege96d.cb	|- ( ph -> C R B )

Assertion

Ref	Expression
frege96d	|- ( ph -> A ( t+ ` R ) B )

Proof of Theorem frege96d

Step	Hyp	Ref	Expression
1		frege96d.a	. . 3 |- ( ph -> A e. _V )
2		frege96d.b	. . 3 |- ( ph -> B e. _V )
3		frege96d.c	. . 3 |- ( ph -> C e. _V )
4		frege96d.ac	. . 3 |- ( ph -> A ( t+ ` R ) C )
5		frege96d.cb	. . 3 |- ( ph -> C R B )
6		brcogw 5290	. . 3 |- ( ( ( A e. _V /\ B e. _V /\ C e. _V ) /\ ( A ( t+ ` R ) C /\ C R B ) ) -> A ( R o. ( t+ ` R ) ) B )
7	1, 2, 3, 4, 5, 6	syl32anc 1334	. 2 |- ( ph -> A ( R o. ( t+ ` R ) ) B )
8		frege96d.r	. . . . 5 |- ( ph -> R e. _V )
9		trclfvlb 13749	. . . . 5 |- ( R e. _V -> R C_ ( t+ ` R ) )
10		coss1 5277	. . . . 5 |- ( R C_ ( t+ ` R ) -> ( R o. ( t+ ` R ) ) C_ ( ( t+ ` R ) o. ( t+ ` R ) ) )
11	8, 9, 10	3syl 18	. . . 4 |- ( ph -> ( R o. ( t+ ` R ) ) C_ ( ( t+ ` R ) o. ( t+ ` R ) ) )
12		trclfvcotrg 13757	. . . 4 |- ( ( t+ ` R ) o. ( t+ ` R ) ) C_ ( t+ ` R )
13	11, 12	syl6ss 3615	. . 3 |- ( ph -> ( R o. ( t+ ` R ) ) C_ ( t+ ` R ) )
14	13	ssbrd 4696	. 2 |- ( ph -> ( A ( R o. ( t+ ` R ) ) B -> A ( t+ ` R ) B ) )
15	7, 14	mpd 15	1 |- ( ph -> A ( t+ ` R ) B )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   o. ccom 5118   ` cfv 5888   t+ctcl 13724

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-trcl 13726

This theorem is referenced by:  frege87d  38042  frege102d  38046  frege129d  38055