Theorem refimssco 37913

Description: Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.)

Assertion

Ref	Expression
refimssco	|- ( ( _I |` ( dom A u. ran A ) ) C_ A -> `' A C_ `' ( A o. A ) )

Proof of Theorem refimssco

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4657	. . . . . . . . . . 11 |- ( z = x -> ( x A z <-> x A x ) )
2		breq1 4656	. . . . . . . . . . 11 |- ( z = x -> ( z A y <-> x A y ) )
3	1, 2	anbi12d 747	. . . . . . . . . 10 |- ( z = x -> ( ( x A z /\ z A y ) <-> ( x A x /\ x A y ) ) )
4	3	biimprd 238	. . . . . . . . 9 |- ( z = x -> ( ( x A x /\ x A y ) -> ( x A z /\ z A y ) ) )
5	4	spimev 2259	. . . . . . . 8 |- ( ( x A x /\ x A y ) -> E. z ( x A z /\ z A y ) )
6	5	ex 450	. . . . . . 7 |- ( x A x -> ( x A y -> E. z ( x A z /\ z A y ) ) )
7	6	adantr 481	. . . . . 6 |- ( ( x A x /\ y A y ) -> ( x A y -> E. z ( x A z /\ z A y ) ) )
8	7	com12 32	. . . . 5 |- ( x A y -> ( ( x A x /\ y A y ) -> E. z ( x A z /\ z A y ) ) )
9	8	a2i 14	. . . 4 |- ( ( x A y -> ( x A x /\ y A y ) ) -> ( x A y -> E. z ( x A z /\ z A y ) ) )
10		19.37v 1910	. . . 4 |- ( E. z ( x A y -> ( x A z /\ z A y ) ) <-> ( x A y -> E. z ( x A z /\ z A y ) ) )
11	9, 10	sylibr 224	. . 3 |- ( ( x A y -> ( x A x /\ y A y ) ) -> E. z ( x A y -> ( x A z /\ z A y ) ) )
12	11	2alimi 1740	. 2 |- ( A. x A. y ( x A y -> ( x A x /\ y A y ) ) -> A. x A. y E. z ( x A y -> ( x A z /\ z A y ) ) )
13		reflexg 37911	. 2 |- ( ( _I |` ( dom A u. ran A ) ) C_ A <-> A. x A. y ( x A y -> ( x A x /\ y A y ) ) )
14		cnvssco 37912	. 2 |- ( `' A C_ `' ( A o. A ) <-> A. x A. y E. z ( x A y -> ( x A z /\ z A y ) ) )
15	12, 13, 14	3imtr4i 281	1 |- ( ( _I |` ( dom A u. ran A ) ) C_ A -> `' A C_ `' ( A o. A ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   u. cun 3572   C_ wss 3574   class class class wbr 4653   _I cid 5023   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118

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-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-id 5024  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)