Theorem xpider 7818

Description: A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
xpider	|- ( A X. A ) Er A

Proof of Theorem xpider

Step	Hyp	Ref	Expression
1		relxp 5227	. 2 |- Rel ( A X. A )
2		dmxpid 5345	. 2 |- dom ( A X. A ) = A
3		cnvxp 5551	. . 3 |- `' ( A X. A ) = ( A X. A )
4		xpidtr 5518	. . 3 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )
5		uneq1 3760	. . . 4 |- ( `' ( A X. A ) = ( A X. A ) -> ( `' ( A X. A ) u. ( A X. A ) ) = ( ( A X. A ) u. ( A X. A ) ) )
6		unss2 3784	. . . 4 |- ( ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A ) -> ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( `' ( A X. A ) u. ( A X. A ) ) )
7		unidm 3756	. . . . 5 |- ( ( A X. A ) u. ( A X. A ) ) = ( A X. A )
8		eqtr 2641	. . . . . 6 |- ( ( ( `' ( A X. A ) u. ( A X. A ) ) = ( ( A X. A ) u. ( A X. A ) ) /\ ( ( A X. A ) u. ( A X. A ) ) = ( A X. A ) ) -> ( `' ( A X. A ) u. ( A X. A ) ) = ( A X. A ) )
9		sseq2 3627	. . . . . . 7 |- ( ( `' ( A X. A ) u. ( A X. A ) ) = ( A X. A ) -> ( ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( `' ( A X. A ) u. ( A X. A ) ) <-> ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( A X. A ) ) )
10	9	biimpd 219	. . . . . 6 |- ( ( `' ( A X. A ) u. ( A X. A ) ) = ( A X. A ) -> ( ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( `' ( A X. A ) u. ( A X. A ) ) -> ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( A X. A ) ) )
11	8, 10	syl 17	. . . . 5 |- ( ( ( `' ( A X. A ) u. ( A X. A ) ) = ( ( A X. A ) u. ( A X. A ) ) /\ ( ( A X. A ) u. ( A X. A ) ) = ( A X. A ) ) -> ( ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( `' ( A X. A ) u. ( A X. A ) ) -> ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( A X. A ) ) )
12	7, 11	mpan2 707	. . . 4 |- ( ( `' ( A X. A ) u. ( A X. A ) ) = ( ( A X. A ) u. ( A X. A ) ) -> ( ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( `' ( A X. A ) u. ( A X. A ) ) -> ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( A X. A ) ) )
13	5, 6, 12	syl2im 40	. . 3 |- ( `' ( A X. A ) = ( A X. A ) -> ( ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A ) -> ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( A X. A ) ) )
14	3, 4, 13	mp2 9	. 2 |- ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( A X. A )
15		df-er 7742	. 2 |- ( ( A X. A ) Er A <-> ( Rel ( A X. A ) /\ dom ( A X. A ) = A /\ ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( A X. A ) ) )
16	1, 2, 14, 15	mpbir3an 1244	1 |- ( A X. A ) Er A

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   u. cun 3572   C_ wss 3574   X. cxp 5112   `'ccnv 5113   dom cdm 5114   o. ccom 5118   Rel wrel 5119   Er wer 7739

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-er 7742

This theorem is referenced by:  riiner  7820  efglem  18129