Theorem xpima2m 5230

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 4782	. . . 4 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) |` C )
2		df-res 4781	. . . . 5 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
3	2	rneqi 5005	. . . 4 |- ran ( ( A X. B ) |` C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
4		inxp 4909	. . . . 5 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
5	4	rneqi 5005	. . . 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 2263	. . 3 |- ( ( A X. B ) " C ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
7		rnxpm 5212	. . 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 2283	. 2 |- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = ( B i^i _V ) )
9		inv1 3559	. 2 |- ( B i^i _V ) = B
10	8, 9	eqtrdi 2287	1 |- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = B )

Syntax hints:   -> wi 4   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   i^i cin 3219   X. cxp 4767   ran crn 4770   |` cres 4771   "cima 4772

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782

This theorem is referenced by:  xpimasn  5231