Theorem brco3f1o 38331

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

Hypotheses

Ref	Expression
brco3f1o.c	|- ( ph -> C : Y -1-1-onto-> Z )
brco3f1o.d	|- ( ph -> D : X -1-1-onto-> Y )
brco3f1o.e	|- ( ph -> E : W -1-1-onto-> X )
brco3f1o.r	|- ( ph -> A ( C o. ( D o. E ) ) B )

Assertion

Ref	Expression
brco3f1o	|- ( ph -> ( ( `' C ` B ) C B /\ ( `' D ` ( `' C ` B ) ) D ( `' C ` B ) /\ A E ( `' D ` ( `' C ` B ) ) ) )

Proof of Theorem brco3f1o

Step	Hyp	Ref	Expression
1		brco3f1o.e	. . . 4 |- ( ph -> E : W -1-1-onto-> X )
2		f1ocnv 6149	. . . 4 |- ( E : W -1-1-onto-> X -> `' E : X -1-1-onto-> W )
3		f1ofn 6138	. . . 4 |- ( `' E : X -1-1-onto-> W -> `' E Fn X )
4	1, 2, 3	3syl 18	. . 3 |- ( ph -> `' E Fn X )
5		brco3f1o.d	. . . 4 |- ( ph -> D : X -1-1-onto-> Y )
6		f1ocnv 6149	. . . 4 |- ( D : X -1-1-onto-> Y -> `' D : Y -1-1-onto-> X )
7		f1of 6137	. . . 4 |- ( `' D : Y -1-1-onto-> X -> `' D : Y --> X )
8	5, 6, 7	3syl 18	. . 3 |- ( ph -> `' D : Y --> X )
9		brco3f1o.c	. . . 4 |- ( ph -> C : Y -1-1-onto-> Z )
10		f1ocnv 6149	. . . 4 |- ( C : Y -1-1-onto-> Z -> `' C : Z -1-1-onto-> Y )
11		f1of 6137	. . . 4 |- ( `' C : Z -1-1-onto-> Y -> `' C : Z --> Y )
12	9, 10, 11	3syl 18	. . 3 |- ( ph -> `' C : Z --> Y )
13		brco3f1o.r	. . . 4 |- ( ph -> A ( C o. ( D o. E ) ) B )
14		relco 5633	. . . . . 6 |- Rel ( ( C o. D ) o. E )
15	14	relbrcnv 5506	. . . . 5 |- ( B `' ( ( C o. D ) o. E ) A <-> A ( ( C o. D ) o. E ) B )
16		cnvco 5308	. . . . . . 7 |- `' ( ( C o. D ) o. E ) = ( `' E o. `' ( C o. D ) )
17		cnvco 5308	. . . . . . . 8 |- `' ( C o. D ) = ( `' D o. `' C )
18	17	coeq2i 5282	. . . . . . 7 |- ( `' E o. `' ( C o. D ) ) = ( `' E o. ( `' D o. `' C ) )
19	16, 18	eqtri 2644	. . . . . 6 |- `' ( ( C o. D ) o. E ) = ( `' E o. ( `' D o. `' C ) )
20	19	breqi 4659	. . . . 5 |- ( B `' ( ( C o. D ) o. E ) A <-> B ( `' E o. ( `' D o. `' C ) ) A )
21		coass 5654	. . . . . 6 |- ( ( C o. D ) o. E ) = ( C o. ( D o. E ) )
22	21	breqi 4659	. . . . 5 |- ( A ( ( C o. D ) o. E ) B <-> A ( C o. ( D o. E ) ) B )
23	15, 20, 22	3bitr3ri 291	. . . 4 |- ( A ( C o. ( D o. E ) ) B <-> B ( `' E o. ( `' D o. `' C ) ) A )
24	13, 23	sylib 208	. . 3 |- ( ph -> B ( `' E o. ( `' D o. `' C ) ) A )
25	4, 8, 12, 24	brcofffn 38329	. 2 |- ( ph -> ( B `' C ( `' C ` B ) /\ ( `' C ` B ) `' D ( `' D ` ( `' C ` B ) ) /\ ( `' D ` ( `' C ` B ) ) `' E A ) )
26		f1orel 6140	. . . 4 |- ( C : Y -1-1-onto-> Z -> Rel C )
27		relbrcnvg 5504	. . . 4 |- ( Rel C -> ( B `' C ( `' C ` B ) <-> ( `' C ` B ) C B ) )
28	9, 26, 27	3syl 18	. . 3 |- ( ph -> ( B `' C ( `' C ` B ) <-> ( `' C ` B ) C B ) )
29		f1orel 6140	. . . 4 |- ( D : X -1-1-onto-> Y -> Rel D )
30		relbrcnvg 5504	. . . 4 |- ( Rel D -> ( ( `' C ` B ) `' D ( `' D ` ( `' C ` B ) ) <-> ( `' D ` ( `' C ` B ) ) D ( `' C ` B ) ) )
31	5, 29, 30	3syl 18	. . 3 |- ( ph -> ( ( `' C ` B ) `' D ( `' D ` ( `' C ` B ) ) <-> ( `' D ` ( `' C ` B ) ) D ( `' C ` B ) ) )
32		f1orel 6140	. . . 4 |- ( E : W -1-1-onto-> X -> Rel E )
33		relbrcnvg 5504	. . . 4 |- ( Rel E -> ( ( `' D ` ( `' C ` B ) ) `' E A <-> A E ( `' D ` ( `' C ` B ) ) ) )
34	1, 32, 33	3syl 18	. . 3 |- ( ph -> ( ( `' D ` ( `' C ` B ) ) `' E A <-> A E ( `' D ` ( `' C ` B ) ) ) )
35	28, 31, 34	3anbi123d 1399	. 2 |- ( ph -> ( ( B `' C ( `' C ` B ) /\ ( `' C ` B ) `' D ( `' D ` ( `' C ` B ) ) /\ ( `' D ` ( `' C ` B ) ) `' E A ) <-> ( ( `' C ` B ) C B /\ ( `' D ` ( `' C ` B ) ) D ( `' C ` B ) /\ A E ( `' D ` ( `' C ` B ) ) ) ) )
36	25, 35	mpbid 222	1 |- ( ph -> ( ( `' C ` B ) C B /\ ( `' D ` ( `' C ` B ) ) D ( `' C ` B ) /\ A E ( `' D ` ( `' C ` B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   class class class wbr 4653   `'ccnv 5113   o. ccom 5118   Rel wrel 5119   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` 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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by: (None)