Theorem trrelsuperrel2dg 37963

Description: Concrete construction of a superclass of relation R which is a transitive relation. (Contributed by RP, 20-Jul-2020.)

Hypothesis

Ref	Expression
trrelsuperrel2dg.s	|- ( ph -> S = ( R u. ( dom R X. ran R ) ) )

Assertion

Ref	Expression
trrelsuperrel2dg	|- ( ph -> ( R C_ S /\ ( S o. S ) C_ S ) )

Proof of Theorem trrelsuperrel2dg

Step	Hyp	Ref	Expression
1		ssun1 3776	. . 3 |- R C_ ( R u. ( dom R X. ran R ) )
2		trrelsuperrel2dg.s	. . 3 |- ( ph -> S = ( R u. ( dom R X. ran R ) ) )
3	1, 2	syl5sseqr 3654	. 2 |- ( ph -> R C_ S )
4		xptrrel 13719	. . . . 5 |- ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) C_ ( dom R X. ran R )
5		ssun2 3777	. . . . 5 |- ( dom R X. ran R ) C_ ( R u. ( dom R X. ran R ) )
6	4, 5	sstri 3612	. . . 4 |- ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) C_ ( R u. ( dom R X. ran R ) )
7	6	a1i 11	. . 3 |- ( ph -> ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) C_ ( R u. ( dom R X. ran R ) ) )
8	2, 2	coeq12d 5286	. . . 4 |- ( ph -> ( S o. S ) = ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) )
9		coundir 5637	. . . . . 6 |- ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) = ( ( R o. ( R u. ( dom R X. ran R ) ) ) u. ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) )
10		relcnv 5503	. . . . . . 7 |- Rel `' `' R
11		cocnvcnv1 5646	. . . . . . . . 9 |- ( `' `' R o. ( R u. ( dom R X. ran R ) ) ) = ( R o. ( R u. ( dom R X. ran R ) ) )
12		relssdmrn 5656	. . . . . . . . . . 11 |- ( Rel `' `' R -> `' `' R C_ ( dom `' `' R X. ran `' `' R ) )
13		dmcnvcnv 5348	. . . . . . . . . . . 12 |- dom `' `' R = dom R
14		rncnvcnv 5349	. . . . . . . . . . . 12 |- ran `' `' R = ran R
15	13, 14	xpeq12i 5137	. . . . . . . . . . 11 |- ( dom `' `' R X. ran `' `' R ) = ( dom R X. ran R )
16	12, 15	syl6sseq 3651	. . . . . . . . . 10 |- ( Rel `' `' R -> `' `' R C_ ( dom R X. ran R ) )
17		coss1 5277	. . . . . . . . . 10 |- ( `' `' R C_ ( dom R X. ran R ) -> ( `' `' R o. ( R u. ( dom R X. ran R ) ) ) C_ ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) )
18	16, 17	syl 17	. . . . . . . . 9 |- ( Rel `' `' R -> ( `' `' R o. ( R u. ( dom R X. ran R ) ) ) C_ ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) )
19	11, 18	syl5eqssr 3650	. . . . . . . 8 |- ( Rel `' `' R -> ( R o. ( R u. ( dom R X. ran R ) ) ) C_ ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) )
20		ssequn1 3783	. . . . . . . 8 |- ( ( R o. ( R u. ( dom R X. ran R ) ) ) C_ ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) <-> ( ( R o. ( R u. ( dom R X. ran R ) ) ) u. ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) ) = ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) )
21	19, 20	sylib 208	. . . . . . 7 |- ( Rel `' `' R -> ( ( R o. ( R u. ( dom R X. ran R ) ) ) u. ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) ) = ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) )
22	10, 21	ax-mp 5	. . . . . 6 |- ( ( R o. ( R u. ( dom R X. ran R ) ) ) u. ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) ) = ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) )
23	9, 22	eqtri 2644	. . . . 5 |- ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) = ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) )
24		coundi 5636	. . . . . 6 |- ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) = ( ( ( dom R X. ran R ) o. R ) u. ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
25		cocnvcnv2 5647	. . . . . . . . 9 |- ( ( dom R X. ran R ) o. `' `' R ) = ( ( dom R X. ran R ) o. R )
26		coss2 5278	. . . . . . . . . 10 |- ( `' `' R C_ ( dom R X. ran R ) -> ( ( dom R X. ran R ) o. `' `' R ) C_ ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
27	16, 26	syl 17	. . . . . . . . 9 |- ( Rel `' `' R -> ( ( dom R X. ran R ) o. `' `' R ) C_ ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
28	25, 27	syl5eqssr 3650	. . . . . . . 8 |- ( Rel `' `' R -> ( ( dom R X. ran R ) o. R ) C_ ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
29		ssequn1 3783	. . . . . . . 8 |- ( ( ( dom R X. ran R ) o. R ) C_ ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) <-> ( ( ( dom R X. ran R ) o. R ) u. ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) ) = ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
30	28, 29	sylib 208	. . . . . . 7 |- ( Rel `' `' R -> ( ( ( dom R X. ran R ) o. R ) u. ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) ) = ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
31	10, 30	ax-mp 5	. . . . . 6 |- ( ( ( dom R X. ran R ) o. R ) u. ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) ) = ( ( dom R X. ran R ) o. ( dom R X. ran R ) )
32	24, 31	eqtri 2644	. . . . 5 |- ( ( dom R X. ran R ) o. ( R u. ( dom R X. ran R ) ) ) = ( ( dom R X. ran R ) o. ( dom R X. ran R ) )
33	23, 32	eqtri 2644	. . . 4 |- ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) = ( ( dom R X. ran R ) o. ( dom R X. ran R ) )
34	8, 33	syl6eq 2672	. . 3 |- ( ph -> ( S o. S ) = ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
35	7, 34, 2	3sstr4d 3648	. 2 |- ( ph -> ( S o. S ) C_ S )
36	3, 35	jca 554	1 |- ( ph -> ( R C_ S /\ ( S o. S ) C_ S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   u. cun 3572   C_ wss 3574   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   o. ccom 5118   Rel wrel 5119

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-ne 2795  df-ral 2917  df-rex 2918  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-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by: (None)