Theorem scprefat2 14381

Description: A square cross product (A X. A) is reflexive.

Assertion

Ref	Expression
scprefat2	|- ( _I |` A) C_ (A X. A)

Proof of Theorem scprefat2

Step	Hyp	Ref	Expression
1		fldsqcp2 14378	. 2 |- U.U.(A X. A) = A
2		reseq2 4219	. . 3 |- (U.U.(A X. A) = A -> ( _I |` U.U.(A X. A)) = ( _I |` A))
3		scprefat 14380	. . . . 5 |- ( _I |` U.U.(A X. A)) C_ (A X. A)
4		sseq1 2637	. . . . 5 |- (( _I |` A) = ( _I |` U.U.(A X. A)) -> (( _I |` A) C_ (A X. A) <-> ( _I |` U.U.(A X. A)) C_ (A X. A)))
5	3, 4	mpbiri 211	. . . 4 |- (( _I |` A) = ( _I |` U.U.(A X. A)) -> ( _I |` A) C_ (A X. A))
6	5	eqcoms 1887	. . 3 |- (( _I |` U.U.(A X. A)) = ( _I |` A) -> ( _I |` A) C_ (A X. A))
7	2, 6	syl 12	. 2 |- (U.U.(A X. A) = A -> ( _I |` A) C_ (A X. A))
8	1, 7	ax-mp 7	1 |- ( _I |` A) C_ (A X. A)

Syntax hints:   = wceq 1298   C_ wss 2593  U.cuni 3177   _I cid 3582   X. cxp 3984   |` cres 3988

This theorem is referenced by:  residcp 14392

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

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