Theorem xpider 6692

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 4783	. 2 |- Rel ( A X. A )
2		dmxpid 4898	. 2 |- dom ( A X. A ) = A
3		cnvxp 5100	. . 3 |- `' ( A X. A ) = ( A X. A )
4		xpidtr 5072	. . 3 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )
5		uneq1 3319	. . . 4 |- ( `' ( A X. A ) = ( A X. A ) -> ( `' ( A X. A ) u. ( A X. A ) ) = ( ( A X. A ) u. ( A X. A ) ) )
6		unss2 3343	. . . 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 3315	. . . . 5 |- ( ( A X. A ) u. ( A X. A ) ) = ( A X. A )
8		eqtr 2222	. . . . . 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 3216	. . . . . . 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 144	. . . . . 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 425	. . . 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 6619	. 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 1181	1 |- ( A X. A ) Er A

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   u. cun 3163   C_ wss 3165   X. cxp 4672   `'ccnv 4673   dom cdm 4674   o. ccom 4678   Rel wrel 4679   Er wer 6616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-er 6619

This theorem is referenced by: (None)