Theorem sltval 31800

Description: The value of the surreal less than relationship. (Contributed by Scott Fenton, 14-Jun-2011.)

Assertion

Ref	Expression
sltval	|- ( ( A e. No /\ B e. No ) -> ( A <s B <-> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )

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

Proof of Theorem sltval

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . 5 |- ( f = A -> ( f e. No <-> A e. No ) )
2	1	anbi1d 741	. . . 4 |- ( f = A -> ( ( f e. No /\ g e. No ) <-> ( A e. No /\ g e. No ) ) )
3		fveq1 6190	. . . . . . . 8 |- ( f = A -> ( f ` y ) = ( A ` y ) )
4	3	eqeq1d 2624	. . . . . . 7 |- ( f = A -> ( ( f ` y ) = ( g ` y ) <-> ( A ` y ) = ( g ` y ) ) )
5	4	ralbidv 2986	. . . . . 6 |- ( f = A -> ( A. y e. x ( f ` y ) = ( g ` y ) <-> A. y e. x ( A ` y ) = ( g ` y ) ) )
6		fveq1 6190	. . . . . . 7 |- ( f = A -> ( f ` x ) = ( A ` x ) )
7	6	breq1d 4663	. . . . . 6 |- ( f = A -> ( ( f ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) <-> ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) )
8	5, 7	anbi12d 747	. . . . 5 |- ( f = A -> ( ( A. y e. x ( f ` y ) = ( g ` y ) /\ ( f ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) <-> ( A. y e. x ( A ` y ) = ( g ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) ) )
9	8	rexbidv 3052	. . . 4 |- ( f = A -> ( E. x e. On ( A. y e. x ( f ` y ) = ( g ` y ) /\ ( f ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) <-> E. x e. On ( A. y e. x ( A ` y ) = ( g ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) ) )
10	2, 9	anbi12d 747	. . 3 |- ( f = A -> ( ( ( f e. No /\ g e. No ) /\ E. x e. On ( A. y e. x ( f ` y ) = ( g ` y ) /\ ( f ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) ) <-> ( ( A e. No /\ g e. No ) /\ E. x e. On ( A. y e. x ( A ` y ) = ( g ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) ) ) )
11		eleq1 2689	. . . . 5 |- ( g = B -> ( g e. No <-> B e. No ) )
12	11	anbi2d 740	. . . 4 |- ( g = B -> ( ( A e. No /\ g e. No ) <-> ( A e. No /\ B e. No ) ) )
13		fveq1 6190	. . . . . . . 8 |- ( g = B -> ( g ` y ) = ( B ` y ) )
14	13	eqeq2d 2632	. . . . . . 7 |- ( g = B -> ( ( A ` y ) = ( g ` y ) <-> ( A ` y ) = ( B ` y ) ) )
15	14	ralbidv 2986	. . . . . 6 |- ( g = B -> ( A. y e. x ( A ` y ) = ( g ` y ) <-> A. y e. x ( A ` y ) = ( B ` y ) ) )
16		fveq1 6190	. . . . . . 7 |- ( g = B -> ( g ` x ) = ( B ` x ) )
17	16	breq2d 4665	. . . . . 6 |- ( g = B -> ( ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) <-> ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) )
18	15, 17	anbi12d 747	. . . . 5 |- ( g = B -> ( ( A. y e. x ( A ` y ) = ( g ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) <-> ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )
19	18	rexbidv 3052	. . . 4 |- ( g = B -> ( E. x e. On ( A. y e. x ( A ` y ) = ( g ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) <-> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )
20	12, 19	anbi12d 747	. . 3 |- ( g = B -> ( ( ( A e. No /\ g e. No ) /\ E. x e. On ( A. y e. x ( A ` y ) = ( g ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) ) <-> ( ( A e. No /\ B e. No ) /\ E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) ) )
21		df-slt 31797	. . 3 |- <s = { <. f , g >. | ( ( f e. No /\ g e. No ) /\ E. x e. On ( A. y e. x ( f ` y ) = ( g ` y ) /\ ( f ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( g ` x ) ) ) }
22	10, 20, 21	brabg 4994	. 2 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> ( ( A e. No /\ B e. No ) /\ E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) ) )
23	22	bianabs 924	1 |- ( ( A e. No /\ B e. No ) -> ( A <s B <-> E. x e. On ( A. y e. x ( A ` y ) = ( B ` y ) /\ ( A ` x ) { <. 1o , (/) >. , <. 1o , 2o >. , <. (/) , 2o >. } ( B ` x ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   (/)c0 3915   {ctp 4181   <.cop 4183   class class class wbr 4653   Oncon0 5723   ` cfv 5888   1oc1o 7553   2oc2o 7554   Nocsur 31793   <scslt 31794

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-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  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-iota 5851  df-fv 5896  df-slt 31797

This theorem is referenced by:  sltval2  31809  sltres  31815  nolesgn2o  31824  nodense  31842  nolt02o  31845  nosupbnd2lem1  31861