Theorem dmscut 31918

Description: The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
dmscut	|- dom |s = < <s

Proof of Theorem dmscut

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

Step	Hyp	Ref	Expression
1		dmoprab 6741	. 2 |- dom { <. <. a , b >. , c >. | ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) } = { <. a , b >. | E. c ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) }
2		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 ) } ) ) )
3		df-mpt2 6655	. . . 4 |- ( 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 ) } ) ) ) = { <. <. a , b >. , c >. | ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) }
4	2, 3	eqtri 2644	. . 3 |- |s = { <. <. a , b >. , c >. | ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) }
5	4	dmeqi 5325	. 2 |- dom |s = dom { <. <. a , b >. , c >. | ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) }
6		df-sslt 31897	. . . . 5 |- < <s = { <. a , b >. | ( a C_ No /\ b C_ No /\ A. x e. a A. y e. b x <s y ) }
7	6	relopabi 5245	. . . 4 |- Rel < <s
8		19.42v 1918	. . . . . 6 |- ( E. c ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) <-> ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ E. c c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) )
9		ssltss1 31903	. . . . . . . . 9 |- ( a < <s b -> a C_ No )
10		vex 3203	. . . . . . . . . 10 |- a e. _V
11	10	elpw 4164	. . . . . . . . 9 |- ( a e. ~P No <-> a C_ No )
12	9, 11	sylibr 224	. . . . . . . 8 |- ( a < <s b -> a e. ~P No )
13	12	pm4.71ri 665	. . . . . . 7 |- ( a < <s b <-> ( a e. ~P No /\ a < <s b ) )
14		vex 3203	. . . . . . . . . 10 |- b e. _V
15	10, 14	elimasn 5490	. . . . . . . . 9 |- ( b e. ( < <s " { a } ) <-> <. a , b >. e. < <s )
16		df-br 4654	. . . . . . . . 9 |- ( a < <s b <-> <. a , b >. e. < <s )
17	15, 16	bitr4i 267	. . . . . . . 8 |- ( b e. ( < <s " { a } ) <-> a < <s b )
18	17	anbi2i 730	. . . . . . 7 |- ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) <-> ( a e. ~P No /\ a < <s b ) )
19		riotaex 6615	. . . . . . . . 9 |- ( 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
20		isset 3207	. . . . . . . . 9 |- ( ( 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 <-> E. c c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) )
21	19, 20	mpbi 220	. . . . . . . 8 |- E. c c = ( 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	21	biantru 526	. . . . . . 7 |- ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) <-> ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ E. c c = ( 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	13, 18, 22	3bitr2i 288	. . . . . 6 |- ( a < <s b <-> ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ E. c c = ( 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	8, 23, 16	3bitr2ri 289	. . . . 5 |- ( <. a , b >. e. < <s <-> E. c ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) )
25	24	a1i 11	. . . 4 |- ( T. -> ( <. a , b >. e. < <s <-> E. c ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) ) )
26	7, 25	opabbi2dv 5271	. . 3 |- ( T. -> < <s = { <. a , b >. | E. c ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) } )
27	26	trud 1493	. 2 |- < <s = { <. a , b >. | E. c ( ( a e. ~P No /\ b e. ( < <s " { a } ) ) /\ c = ( iota_ x e. { y e. No | ( a < <s { y } /\ { y } < <s b ) } ( bday ` x ) = |^| ( bday " { y e. No | ( a < <s { y } /\ { y } < <s b ) } ) ) ) }
28	1, 5, 27	3eqtr4i 2654	1 |- dom |s = < <s

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   E.wex 1704   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   <.cop 4183   |^|cint 4475   class class class wbr 4653   {copab 4712   dom cdm 5114   "cima 5117   ` cfv 5888   iota_crio 6610   {coprab 6651   |-> cmpt2 6652   Nocsur 31793   <scslt 31794   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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-riota 6611  df-oprab 6654  df-mpt2 6655  df-sslt 31897  df-scut 31899

This theorem is referenced by:  scutf  31919  madeval2  31936