Theorem xpima2m 5182

Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
xpima2m	|- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = B )

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem xpima2m

Step	Hyp	Ref	Expression
1		df-ima 4736	. . . 4 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) |` C )
2		df-res 4735	. . . . 5 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
3	2	rneqi 4958	. . . 4 |- ran ( ( A X. B ) |` C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
4		inxp 4862	. . . . 5 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
5	4	rneqi 4958	. . . 4 |- ran ( ( A X. B ) i^i ( C X. _V ) ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
6	1, 3, 5	3eqtri 2254	. . 3 |- ( ( A X. B ) " C ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
7		rnxpm 5164	. . 3 |- ( E. x x e. ( A i^i C ) -> ran ( ( A i^i C ) X. ( B i^i _V ) ) = ( B i^i _V ) )
8	6, 7	eqtrid 2274	. 2 |- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = ( B i^i _V ) )
9		inv1 3529	. 2 |- ( B i^i _V ) = B
10	8, 9	eqtrdi 2278	1 |- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = B )

Syntax hints:   -> wi 4   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2800   i^i cin 3197   X. cxp 4721   ran crn 4724   |` cres 4725   "cima 4726

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736

This theorem is referenced by:  xpimasn  5183