Theorem itg2l 23496

Description: Elementhood in the set L of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Hypothesis

Ref	Expression
itg2val.1	|- L = { x | E. g e. dom S.1 ( g oR <_ F /\ x = ( S.1 ` g ) ) }

Assertion

Ref	Expression
itg2l	|- ( A e. L <-> E. g e. dom S.1 ( g oR <_ F /\ A = ( S.1 ` g ) ) )

Distinct variable groups:   x, g, A   g, F, x

Allowed substitution hints:   L( x, g)

Proof of Theorem itg2l

Step	Hyp	Ref	Expression
1		itg2val.1	. . 3 |- L = { x | E. g e. dom S.1 ( g oR <_ F /\ x = ( S.1 ` g ) ) }
2	1	eleq2i 2693	. 2 |- ( A e. L <-> A e. { x | E. g e. dom S.1 ( g oR <_ F /\ x = ( S.1 ` g ) ) } )
3		simpr 477	. . . . 5 |- ( ( g oR <_ F /\ A = ( S.1 ` g ) ) -> A = ( S.1 ` g ) )
4		fvex 6201	. . . . 5 |- ( S.1 ` g ) e. _V
5	3, 4	syl6eqel 2709	. . . 4 |- ( ( g oR <_ F /\ A = ( S.1 ` g ) ) -> A e. _V )
6	5	rexlimivw 3029	. . 3 |- ( E. g e. dom S.1 ( g oR <_ F /\ A = ( S.1 ` g ) ) -> A e. _V )
7		eqeq1 2626	. . . . 5 |- ( x = A -> ( x = ( S.1 ` g ) <-> A = ( S.1 ` g ) ) )
8	7	anbi2d 740	. . . 4 |- ( x = A -> ( ( g oR <_ F /\ x = ( S.1 ` g ) ) <-> ( g oR <_ F /\ A = ( S.1 ` g ) ) ) )
9	8	rexbidv 3052	. . 3 |- ( x = A -> ( E. g e. dom S.1 ( g oR <_ F /\ x = ( S.1 ` g ) ) <-> E. g e. dom S.1 ( g oR <_ F /\ A = ( S.1 ` g ) ) ) )
10	6, 9	elab3 3358	. 2 |- ( A e. { x | E. g e. dom S.1 ( g oR <_ F /\ x = ( S.1 ` g ) ) } <-> E. g e. dom S.1 ( g oR <_ F /\ A = ( S.1 ` g ) ) )
11	2, 10	bitri 264	1 |- ( A e. L <-> E. g e. dom S.1 ( g oR <_ F /\ A = ( S.1 ` g ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   dom cdm 5114   ` cfv 5888   oRcofr 6896   <_ cle 10075   S.1citg1 23384

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  ax-nul 4789

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-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-fv 5896

This theorem is referenced by:  itg2lr  23497