Theorem xpiderm 6177

Description: A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by Jim Kingdon, 22-Aug-2019.)

Assertion

Ref	Expression
xpiderm	|- ( E. x x e. A -> ( A X. A ) Er A )

Distinct variable group:   x, A

Proof of Theorem xpiderm

Step	Hyp	Ref	Expression
1		relxp 4447	. . 3 |- Rel ( A X. A )
2	1	a1i 9	. 2 |- ( E. x x e. A -> Rel ( A X. A ) )
3		dmxpm 4555	. 2 |- ( E. x x e. A -> dom ( A X. A ) = A )
4		cnvxp 4742	. . . 4 |- `' ( A X. A ) = ( A X. A )
5		xpidtr 4715	. . . 4 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )
6		uneq1 3090	. . . . 5 |- ( `' ( A X. A ) = ( A X. A ) -> ( `' ( A X. A ) u. ( A X. A ) ) = ( ( A X. A ) u. ( A X. A ) ) )
7		unss2 3114	. . . . 5 |- ( ( ( 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 ) ) )
8		unidm 3086	. . . . . 6 |- ( ( A X. A ) u. ( A X. A ) ) = ( A X. A )
9		eqtr 2057	. . . . . . 7 |- ( ( ( `' ( 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 ) )
10		sseq2 2967	. . . . . . . 8 |- ( ( `' ( 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	10	biimpd 132	. . . . . . 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 ) ) )
12	9, 11	syl 14	. . . . . 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 ) 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	8, 12	mpan2 401	. . . . 5 |- ( ( `' ( 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 ) ) )
14	6, 7, 13	syl2im 34	. . . 4 |- ( `' ( 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 ) ) )
15	4, 5, 14	mp2 16	. . 3 |- ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( A X. A )
16	15	a1i 9	. 2 |- ( E. x x e. A -> ( `' ( A X. A ) u. ( ( A X. A ) o. ( A X. A ) ) ) C_ ( A X. A ) )
17		df-er 6106	. 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 ) ) )
18	2, 3, 16, 17	syl3anbrc 1088	1 |- ( E. x x e. A -> ( A X. A ) Er A )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   u. cun 2915   C_ wss 2917   X. cxp 4343   `'ccnv 4344   dom cdm 4345   o. ccom 4349   Rel wrel 4350   Er wer 6103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-er 6106

This theorem is referenced by: (None)