Theorem trrelsuperreldg 37960

Description: Concrete construction of a superclass of relation R which is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)

Hypotheses

Ref	Expression
trrelsuperreldg.r	|- ( ph -> Rel R )
trrelsuperreldg.s	|- ( ph -> S = ( dom R X. ran R ) )

Assertion

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

Proof of Theorem trrelsuperreldg

Step	Hyp	Ref	Expression
1		trrelsuperreldg.r	. . . 4 |- ( ph -> Rel R )
2		relssdmrn 5656	. . . 4 |- ( Rel R -> R C_ ( dom R X. ran R ) )
3	1, 2	syl 17	. . 3 |- ( ph -> R C_ ( dom R X. ran R ) )
4		trrelsuperreldg.s	. . 3 |- ( ph -> S = ( dom R X. ran R ) )
5	3, 4	sseqtr4d 3642	. 2 |- ( ph -> R C_ S )
6		xptrrel 13719	. . . 4 |- ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) C_ ( dom R X. ran R )
7	6	a1i 11	. . 3 |- ( ph -> ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) C_ ( dom R X. ran R ) )
8	4, 4	coeq12d 5286	. . 3 |- ( ph -> ( S o. S ) = ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
9	7, 8, 4	3sstr4d 3648	. 2 |- ( ph -> ( S o. S ) C_ S )
10	5, 9	jca 554	1 |- ( ph -> ( R C_ S /\ ( S o. S ) C_ S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   C_ wss 3574   X. cxp 5112   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)