Theorem iinexg 4824

Description: The existence of a class intersection. x is normally a free-variable parameter in B, which should be read B ( x ). (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
iinexg	|- ( ( A =/= (/) /\ A. x e. A B e. C ) -> |^|_ x e. A B e. _V )

Distinct variable group:   x, A

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

Proof of Theorem iinexg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiin2g 4553	. . 3 |- ( A. x e. A B e. C -> |^|_ x e. A B = |^| { y | E. x e. A y = B } )
2	1	adantl 482	. 2 |- ( ( A =/= (/) /\ A. x e. A B e. C ) -> |^|_ x e. A B = |^| { y | E. x e. A y = B } )
3		elisset 3215	. . . . . . . . 9 |- ( B e. C -> E. y y = B )
4	3	rgenw 2924	. . . . . . . 8 |- A. x e. A ( B e. C -> E. y y = B )
5		r19.2z 4060	. . . . . . . 8 |- ( ( A =/= (/) /\ A. x e. A ( B e. C -> E. y y = B ) ) -> E. x e. A ( B e. C -> E. y y = B ) )
6	4, 5	mpan2 707	. . . . . . 7 |- ( A =/= (/) -> E. x e. A ( B e. C -> E. y y = B ) )
7		r19.35 3084	. . . . . . 7 |- ( E. x e. A ( B e. C -> E. y y = B ) <-> ( A. x e. A B e. C -> E. x e. A E. y y = B ) )
8	6, 7	sylib 208	. . . . . 6 |- ( A =/= (/) -> ( A. x e. A B e. C -> E. x e. A E. y y = B ) )
9	8	imp 445	. . . . 5 |- ( ( A =/= (/) /\ A. x e. A B e. C ) -> E. x e. A E. y y = B )
10		rexcom4 3225	. . . . 5 |- ( E. x e. A E. y y = B <-> E. y E. x e. A y = B )
11	9, 10	sylib 208	. . . 4 |- ( ( A =/= (/) /\ A. x e. A B e. C ) -> E. y E. x e. A y = B )
12		abn0 3954	. . . 4 |- ( { y | E. x e. A y = B } =/= (/) <-> E. y E. x e. A y = B )
13	11, 12	sylibr 224	. . 3 |- ( ( A =/= (/) /\ A. x e. A B e. C ) -> { y | E. x e. A y = B } =/= (/) )
14		intex 4820	. . 3 |- ( { y | E. x e. A y = B } =/= (/) <-> |^| { y | E. x e. A y = B } e. _V )
15	13, 14	sylib 208	. 2 |- ( ( A =/= (/) /\ A. x e. A B e. C ) -> |^| { y | E. x e. A y = B } e. _V )
16	2, 15	eqeltrd 2701	1 |- ( ( A =/= (/) /\ A. x e. A B e. C ) -> |^|_ x e. A B e. _V )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   (/)c0 3915   |^|cint 4475   |^|_ciin 4521

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-int 4476  df-iin 4523

This theorem is referenced by:  fclsval  21812  taylfval  24113  iinexd  39318  smflimlem1  40979  smfliminflem  41036