Theorem brcofffn 38329

Description: Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

Hypotheses

Ref	Expression
brcofffn.c	|- ( ph -> C Fn Z )
brcofffn.d	|- ( ph -> D : Y --> Z )
brcofffn.e	|- ( ph -> E : X --> Y )
brcofffn.r	|- ( ph -> A ( C o. ( D o. E ) ) B )

Assertion

Ref	Expression
brcofffn	|- ( ph -> ( A E ( E ` A ) /\ ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) )

Proof of Theorem brcofffn

Step	Hyp	Ref	Expression
1		brcofffn.c	. . . . 5 |- ( ph -> C Fn Z )
2		brcofffn.d	. . . . 5 |- ( ph -> D : Y --> Z )
3		fnfco 6069	. . . . 5 |- ( ( C Fn Z /\ D : Y --> Z ) -> ( C o. D ) Fn Y )
4	1, 2, 3	syl2anc 693	. . . 4 |- ( ph -> ( C o. D ) Fn Y )
5		brcofffn.e	. . . 4 |- ( ph -> E : X --> Y )
6		brcofffn.r	. . . . 5 |- ( ph -> A ( C o. ( D o. E ) ) B )
7		coass 5654	. . . . . 6 |- ( ( C o. D ) o. E ) = ( C o. ( D o. E ) )
8	7	breqi 4659	. . . . 5 |- ( A ( ( C o. D ) o. E ) B <-> A ( C o. ( D o. E ) ) B )
9	6, 8	sylibr 224	. . . 4 |- ( ph -> A ( ( C o. D ) o. E ) B )
10	4, 5, 9	brcoffn 38328	. . 3 |- ( ph -> ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) )
11	1	adantr 481	. . . . 5 |- ( ( ph /\ ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) ) -> C Fn Z )
12	2	adantr 481	. . . . 5 |- ( ( ph /\ ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) ) -> D : Y --> Z )
13		simprr 796	. . . . 5 |- ( ( ph /\ ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) ) -> ( E ` A ) ( C o. D ) B )
14	11, 12, 13	brcoffn 38328	. . . 4 |- ( ( ph /\ ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) ) -> ( ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) )
15	14	ex 450	. . 3 |- ( ph -> ( ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) -> ( ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) ) )
16	10, 15	jcai 559	. 2 |- ( ph -> ( ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) /\ ( ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) ) )
17		simpll 790	. . 3 |- ( ( ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) /\ ( ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) ) -> A E ( E ` A ) )
18		simprl 794	. . 3 |- ( ( ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) /\ ( ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) ) -> ( E ` A ) D ( D ` ( E ` A ) ) )
19		simprr 796	. . 3 |- ( ( ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) /\ ( ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) ) -> ( D ` ( E ` A ) ) C B )
20	17, 18, 19	3jca 1242	. 2 |- ( ( ( A E ( E ` A ) /\ ( E ` A ) ( C o. D ) B ) /\ ( ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) ) -> ( A E ( E ` A ) /\ ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) )
21	16, 20	syl 17	1 |- ( ph -> ( A E ( E ` A ) /\ ( E ` A ) D ( D ` ( E ` A ) ) /\ ( D ` ( E ` A ) ) C B ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   class class class wbr 4653   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888

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

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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  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-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  brco3f1o  38331  neicvgmex  38415  neicvgel1  38417