Theorem xpiindim 4416

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

Assertion

Ref	Expression
xpiindim	|- (E.y y e. A -> ( C X. |^|_x e. A B) = |^|_x e. A ( C X. B))

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

Allowed substitution hints:   B(x,y)

Proof of Theorem xpiindim

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 4390	. . . . . 6 |- Rel ( C X. B)
2	1	rgenw 2370	. . . . 5 |- A.x e. A Rel ( C X. B)
3		eleq1 2097	. . . . . . 7 |- (x = y -> (x e. A <-> y e. A))
4	3	cbvexv 1792	. . . . . 6 |- (E.x x e. A <-> E.y y e. A)
5		r19.2m 3303	. . . . . 6 |- ((E.x x e. A /\ A.x e. A Rel ( C X. B)) -> E.x e. A Rel ( C X. B))
6	4, 5	sylanbr 269	. . . . 5 |- ((E.y y e. A /\ A.x e. A Rel ( C X. B)) -> E.x e. A Rel ( C X. B))
7	2, 6	mpan2 401	. . . 4 |- (E.y y e. A -> E.x e. A Rel ( C X. B))
8		reliin 4402	. . . 4 |- (E.x e. A Rel ( C X. B) -> Rel |^|_x e. A ( C X. B))
9	7, 8	syl 14	. . 3 |- (E.y y e. A -> Rel |^|_x e. A ( C X. B))
10		relxp 4390	. . 3 |- Rel ( C X. |^|_x e. A B)
11	9, 10	jctil 295	. 2 |- (E.y y e. A -> ( Rel ( C X. |^|_x e. A B) /\ Rel |^|_x e. A ( C X. B)))
12		r19.28mv 3308	. . . . . . 7 |- (E.x x e. A -> (A.x e. A (w e. C /\ z e. B) <-> (w e. C /\ A.x e. A z e. B)))
13	4, 12	sylbir 125	. . . . . 6 |- (E.y y e. A -> (A.x e. A (w e. C /\ z e. B) <-> (w e. C /\ A.x e. A z e. B)))
14	13	bicomd 129	. . . . 5 |- (E.y y e. A -> ((w e. C /\ A.x e. A z e. B) <-> A.x e. A (w e. C /\ z e. B)))
15		vex 2554	. . . . . . 7 |- z e. _V
16		eliin 3653	. . . . . . 7 |- (z e. _V -> (z e. |^|_x e. A B <-> A.x e. A z e. B))
17	15, 16	ax-mp 7	. . . . . 6 |- (z e. |^|_x e. A B <-> A.x e. A z e. B)
18	17	anbi2i 430	. . . . 5 |- ((w e. C /\ z e. |^|_x e. A B) <-> (w e. C /\ A.x e. A z e. B))
19		opelxp 4317	. . . . . 6 |- ( <.w , z >. e. ( C X. B) <-> (w e. C /\ z e. B))
20	19	ralbii 2324	. . . . 5 |- (A.x e. A <.w , z >. e. ( C X. B) <-> A.x e. A (w e. C /\ z e. B))
21	14, 18, 20	3bitr4g 212	. . . 4 |- (E.y y e. A -> ((w e. C /\ z e. |^|_x e. A B) <-> A.x e. A <.w , z >. e. ( C X. B)))
22		opelxp 4317	. . . 4 |- ( <.w , z >. e. ( C X. |^|_x e. A B) <-> (w e. C /\ z e. |^|_x e. A B))
23		vex 2554	. . . . . 6 |- w e. _V
24	23, 15	opex 3957	. . . . 5 |- <.w , z >. e. _V
25		eliin 3653	. . . . 5 |- ( <.w , z >. e. _V -> ( <.w , z >. e. |^|_x e. A ( C X. B) <-> A.x e. A <.w , z >. e. ( C X. B)))
26	24, 25	ax-mp 7	. . . 4 |- ( <.w , z >. e. |^|_x e. A ( C X. B) <-> A.x e. A <.w , z >. e. ( C X. B))
27	21, 22, 26	3bitr4g 212	. . 3 |- (E.y y e. A -> ( <.w , z >. e. ( C X. |^|_x e. A B) <-> <.w , z >. e. |^|_x e. A ( C X. B)))
28	27	eqrelrdv2 4382	. 2 |- ((( Rel ( C X. |^|_x e. A B) /\ Rel |^|_x e. A ( C X. B)) /\ E.y y e. A) -> ( C X. |^|_x e. A B) = |^|_x e. A ( C X. B))
29	11, 28	mpancom 399	1 |- (E.y y e. A -> ( C X. |^|_x e. A B) = |^|_x e. A ( C X. B))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242  E.wex 1378   e. wcel 1390  A.wral 2300  E.wrex 2301   _Vcvv 2551   <.cop 3370   |^|_ciin 3649   X. cxp 4286   Rel wrel 4293

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iin 3651  df-opab 3810  df-xp 4294  df-rel 4295

This theorem is referenced by:  xpriindim  4417