Theorem scutval 31911

Description: The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
scutval	|- ( A < <s B -> ( A |s B ) = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )

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

Proof of Theorem scutval

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssltex1 31901	. . 3 |- ( A < <s B -> A e. _V )
2		ssltss1 31903	. . 3 |- ( A < <s B -> A C_ No )
3	1, 2	elpwd 4167	. 2 |- ( A < <s B -> A e. ~P No )
4		df-br 4654	. . . 4 |- ( A < <s B <-> <. A , B >. e. < <s )
5	4	biimpi 206	. . 3 |- ( A < <s B -> <. A , B >. e. < <s )
6		ssltex2 31902	. . . 4 |- ( A < <s B -> B e. _V )
7		elimasng 5491	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( B e. ( < <s " { A } ) <-> <. A , B >. e. < <s ) )
8	1, 6, 7	syl2anc 693	. . 3 |- ( A < <s B -> ( B e. ( < <s " { A } ) <-> <. A , B >. e. < <s ) )
9	5, 8	mpbird 247	. 2 |- ( A < <s B -> B e. ( < <s " { A } ) )
10		riotaex 6615	. . 3 |- ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) e. _V
11		breq1 4656	. . . . . . 7 |- ( a = A -> ( a < <s { y } <-> A < <s { y } ) )
12		breq2 4657	. . . . . . 7 |- ( b = B -> ( { y } < <s b <-> { y } < <s B ) )
13	11, 12	bi2anan9 917	. . . . . 6 |- ( ( a = A /\ b = B ) -> ( ( a < <s { y } /\ { y } < <s b ) <-> ( A < <s { y } /\ { y } < <s B ) ) )
14	13	rabbidv 3189	. . . . 5 |- ( ( a = A /\ b = B ) -> { y e. No | ( a < <s { y } /\ { y } < <s b ) } = { y e. No | ( A < <s { y } /\ { y } < <s B ) } )
15	14	imaeq2d 5466	. . . . . . 7 |- ( ( a = A /\ b = B ) -> ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) = ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
16	15	inteqd 4480	. . . . . 6 |- ( ( a = A /\ b = B ) -> |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) )
17	16	eqeq2d 2632	. . . . 5 |- ( ( a = A /\ b = B ) -> ( ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) <-> ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )
18	14, 17	riotaeqbidv 6614	. . . 4 |- ( ( a = A /\ b = B ) -> ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )
19		sneq 4187	. . . . 5 |- ( a = A -> { a } = { A } )
20	19	imaeq2d 5466	. . . 4 |- ( a = A -> ( < <s " { a } ) = ( < <s " { A } ) )
21		df-scut 31899	. . . 4 |- |s = ( a e. ~P No , b e. ( < <s " { a } ) |-> ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) )
22	18, 20, 21	ovmpt2x 6789	. . 3 |- ( ( A e. ~P No /\ B e. ( < <s " { A } ) /\ ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) e. _V ) -> ( A |s B ) = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )
23	10, 22	mp3an3 1413	. 2 |- ( ( A e. ~P No /\ B e. ( < <s " { A } ) ) -> ( A |s B ) = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )
24	3, 9, 23	syl2anc 693	1 |- ( A < <s B -> ( A |s B ) = ( iota_ x e. { y e. No | ( A < <s { y } /\ { y } < <s B ) } ( bday ` x ) = |^| ( bday " { y e. No | ( A < <s { y } /\ { y } < <s B ) } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   ~Pcpw 4158   {csn 4177   <.cop 4183   |^|cint 4475   class class class wbr 4653   "cima 5117   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Nocsur 31793   bdaycbday 31795   < <scsslt 31896   |scscut 31898

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-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sslt 31897  df-scut 31899

This theorem is referenced by:  scutcut  31912  scutbday  31913  scutun12  31917  scutf  31919  scutbdaylt  31922