Theorem xpriindim 4859

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 4856	. 2 |- ( C X. ( D i^i |^|_ x e. A B ) ) = ( ( C X. D ) i^i ( C X. |^|_ x e. A B ) )
2		xpiindim 4858	. . 3 |- ( E. y y e. A -> ( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) )
3	2	ineq2d 3405	. 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 2274	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 1395   E.wex 1538   e. wcel 2200   i^i cin 3196   |^|_ciin 3965   X. cxp 4716

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 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-iin 3967  df-opab 4145  df-xp 4724  df-rel 4725

This theorem is referenced by: (None)