Theorem xpima2m 4768

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 4358	. . . 4 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) |` C )
2		df-res 4357	. . . . 5 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
3	2	rneqi 4562	. . . 4 |- ran ( ( A X. B ) |` C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
4		inxp 4470	. . . . 5 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
5	4	rneqi 4562	. . . 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 2064	. . 3 |- ( ( A X. B ) " C ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
7		rnxpm 4752	. . 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	syl5eq 2084	. 2 |- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = ( B i^i _V ) )
9		inv1 3253	. 2 |- ( B i^i _V ) = B
10	8, 9	syl6eq 2088	1 |- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = B )

Syntax hints:   -> wi 4   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557   i^i cin 2916   X. cxp 4343   ran crn 4346   |` cres 4347   "cima 4348

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-dm 4355  df-rn 4356  df-res 4357  df-ima 4358

This theorem is referenced by:  xpimasn  4769