Theorem xpriindim 4760

Description: Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)

Assertion

Ref	Expression
xpriindim	|- ( E. y y e. A -> ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) )

Distinct variable groups:   x, y, A   x, C, y

Allowed substitution hints:   B( x, y)   D( x, y)

Proof of Theorem xpriindim

Step	Hyp	Ref	Expression
1		xpindi 4757	. 2 |- ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i ( C X. |^|_ x e. A B ) )
2		xpiindim 4759	. . 3 |- ( E. y y e. A -> ( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) )
3	2	ineq2d 3336	. 2 |- ( E. y y e. A -> ( ( C X. D ) i^i ( C X. |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) )
4	1, 3	eqtrid 2222	1 |- ( E. y y e. A -> ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) )

Syntax hints:   -> wi 4   = wceq 1353   E.wex 1492   e. wcel 2148   i^i cin 3128   |^|_ciin 3885   X. cxp 4620

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-iin 3887  df-opab 4062  df-xp 4628  df-rel 4629

This theorem is referenced by: (None)