Theorem inixp 33523

Description: Intersection of Cartesian products over the same base set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
inixp	|- ( X_ x e. A B i^i X_ x e. A C ) = X_ x e. A ( B i^i C )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   C( x)

Proof of Theorem inixp

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		an4 865	. . . 4 |- ( ( ( f Fn A /\ A. x e. A ( f ` x ) e. B ) /\ ( f Fn A /\ A. x e. A ( f ` x ) e. C ) ) <-> ( ( f Fn A /\ f Fn A ) /\ ( A. x e. A ( f ` x ) e. B /\ A. x e. A ( f ` x ) e. C ) ) )
2		anidm 676	. . . . 5 |- ( ( f Fn A /\ f Fn A ) <-> f Fn A )
3		r19.26 3064	. . . . . 6 |- ( A. x e. A ( ( f ` x ) e. B /\ ( f ` x ) e. C ) <-> ( A. x e. A ( f ` x ) e. B /\ A. x e. A ( f ` x ) e. C ) )
4		elin 3796	. . . . . . . 8 |- ( ( f ` x ) e. ( B i^i C ) <-> ( ( f ` x ) e. B /\ ( f ` x ) e. C ) )
5	4	bicomi 214	. . . . . . 7 |- ( ( ( f ` x ) e. B /\ ( f ` x ) e. C ) <-> ( f ` x ) e. ( B i^i C ) )
6	5	ralbii 2980	. . . . . 6 |- ( A. x e. A ( ( f ` x ) e. B /\ ( f ` x ) e. C ) <-> A. x e. A ( f ` x ) e. ( B i^i C ) )
7	3, 6	bitr3i 266	. . . . 5 |- ( ( A. x e. A ( f ` x ) e. B /\ A. x e. A ( f ` x ) e. C ) <-> A. x e. A ( f ` x ) e. ( B i^i C ) )
8	2, 7	anbi12i 733	. . . 4 |- ( ( ( f Fn A /\ f Fn A ) /\ ( A. x e. A ( f ` x ) e. B /\ A. x e. A ( f ` x ) e. C ) ) <-> ( f Fn A /\ A. x e. A ( f ` x ) e. ( B i^i C ) ) )
9	1, 8	bitri 264	. . 3 |- ( ( ( f Fn A /\ A. x e. A ( f ` x ) e. B ) /\ ( f Fn A /\ A. x e. A ( f ` x ) e. C ) ) <-> ( f Fn A /\ A. x e. A ( f ` x ) e. ( B i^i C ) ) )
10		vex 3203	. . . . 5 |- f e. _V
11	10	elixp 7915	. . . 4 |- ( f e. X_ x e. A B <-> ( f Fn A /\ A. x e. A ( f ` x ) e. B ) )
12	10	elixp 7915	. . . 4 |- ( f e. X_ x e. A C <-> ( f Fn A /\ A. x e. A ( f ` x ) e. C ) )
13	11, 12	anbi12i 733	. . 3 |- ( ( f e. X_ x e. A B /\ f e. X_ x e. A C ) <-> ( ( f Fn A /\ A. x e. A ( f ` x ) e. B ) /\ ( f Fn A /\ A. x e. A ( f ` x ) e. C ) ) )
14	10	elixp 7915	. . 3 |- ( f e. X_ x e. A ( B i^i C ) <-> ( f Fn A /\ A. x e. A ( f ` x ) e. ( B i^i C ) ) )
15	9, 13, 14	3bitr4i 292	. 2 |- ( ( f e. X_ x e. A B /\ f e. X_ x e. A C ) <-> f e. X_ x e. A ( B i^i C ) )
16	15	ineqri 3806	1 |- ( X_ x e. A B i^i X_ x e. A C ) = X_ x e. A ( B i^i C )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   Fn wfn 5883   ` cfv 5888   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ixp 7909

This theorem is referenced by: (None)