Theorem xpider 6665

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 4772	. 2 |- Rel ( A X. A )
2		dmxpid 4887	. 2 |- dom ( A X. A ) = A
3		cnvxp 5088	. . 3 |- `' ( A X. A ) = ( A X. A )
4		xpidtr 5060	. . 3 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )
5		uneq1 3310	. . . 4 |- ( `' ( A X. A ) = ( A X. A ) -> ( `' ( A X. A ) u. ( A X. A ) ) = ( ( A X. A ) u. ( A X. A ) ) )
6		unss2 3334	. . . 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 3306	. . . . 5 |- ( ( A X. A ) u. ( A X. A ) ) = ( A X. A )
8		eqtr 2214	. . . . . 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 3207	. . . . . . 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 6592	. 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 1364   u. cun 3155   C_ wss 3157   X. cxp 4661   `'ccnv 4662   dom cdm 4663   o. ccom 4667   Rel wrel 4668   Er wer 6589

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-er 6592

This theorem is referenced by: (None)