Theorem xpider 6508

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 4656	. 2 |- Rel ( A X. A )
2		dmxpid 4768	. 2 |- dom ( A X. A ) = A
3		cnvxp 4965	. . 3 |- `' ( A X. A ) = ( A X. A )
4		xpidtr 4937	. . 3 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )
5		uneq1 3228	. . . 4 |- ( `' ( A X. A ) = ( A X. A ) -> ( `' ( A X. A ) u. ( A X. A ) ) = ( ( A X. A ) u. ( A X. A ) ) )
6		unss2 3252	. . . 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 3224	. . . . 5 |- ( ( A X. A ) u. ( A X. A ) ) = ( A X. A )
8		eqtr 2158	. . . . . 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 3126	. . . . . . 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 143	. . . . . 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 14	. . . . 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 422	. . . 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 38	. . 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 16	. 2 |- ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( A X. A )
15		df-er 6437	. 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 1164	1 |- ( A X. A ) Er A

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   u. cun 3074   C_ wss 3076   X. cxp 4545   `'ccnv 4546   dom cdm 4547   o. ccom 4551   Rel wrel 4552   Er wer 6434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-er 6437

This theorem is referenced by: (None)