Theorem scutbday 31913

Description: The birthday of the surreal cut is equal to the minimum birthday in the gap. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem scutbday

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		scutval 31911	. . 3 |- ( A < <s B -> ( A |s B ) = ( iota_ y e. { x e. No | ( A < <s { x } /\ { x } < <s B ) } ( bday ` y ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) ) )
2	1	eqcomd 2628	. 2 |- ( A < <s B -> ( iota_ y e. { x e. No | ( A < <s { x } /\ { x } < <s B ) } ( bday ` y ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) ) = ( A |s B ) )
3		scutcut 31912	. . . 4 |- ( A < <s B -> ( ( A |s B ) e. No /\ A < <s { ( A |s B ) } /\ { ( A |s B ) } < <s B ) )
4		sneq 4187	. . . . . . . 8 |- ( x = ( A |s B ) -> { x } = { ( A |s B ) } )
5	4	breq2d 4665	. . . . . . 7 |- ( x = ( A |s B ) -> ( A < <s { x } <-> A < <s { ( A |s B ) } ) )
6	4	breq1d 4663	. . . . . . 7 |- ( x = ( A |s B ) -> ( { x } < <s B <-> { ( A |s B ) } < <s B ) )
7	5, 6	anbi12d 747	. . . . . 6 |- ( x = ( A |s B ) -> ( ( A < <s { x } /\ { x } < <s B ) <-> ( A < <s { ( A |s B ) } /\ { ( A |s B ) } < <s B ) ) )
8	7	elrab 3363	. . . . 5 |- ( ( A |s B ) e. { x e. No | ( A < <s { x } /\ { x } < <s B ) } <-> ( ( A |s B ) e. No /\ ( A < <s { ( A |s B ) } /\ { ( A |s B ) } < <s B ) ) )
9		3anass 1042	. . . . 5 |- ( ( ( A |s B ) e. No /\ A < <s { ( A |s B ) } /\ { ( A |s B ) } < <s B ) <-> ( ( A |s B ) e. No /\ ( A < <s { ( A |s B ) } /\ { ( A |s B ) } < <s B ) ) )
10	8, 9	bitr4i 267	. . . 4 |- ( ( A |s B ) e. { x e. No | ( A < <s { x } /\ { x } < <s B ) } <-> ( ( A |s B ) e. No /\ A < <s { ( A |s B ) } /\ { ( A |s B ) } < <s B ) )
11	3, 10	sylibr 224	. . 3 |- ( A < <s B -> ( A |s B ) e. { x e. No | ( A < <s { x } /\ { x } < <s B ) } )
12		conway 31910	. . 3 |- ( A < <s B -> E! y e. { x e. No | ( A < <s { x } /\ { x } < <s B ) } ( bday ` y ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) )
13		fveq2 6191	. . . . 5 |- ( y = ( A |s B ) -> ( bday ` y ) = ( bday ` ( A |s B ) ) )
14	13	eqeq1d 2624	. . . 4 |- ( y = ( A |s B ) -> ( ( bday ` y ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) <-> ( bday ` ( A |s B ) ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) ) )
15	14	riota2 6633	. . 3 |- ( ( ( A |s B ) e. { x e. No | ( A < <s { x } /\ { x } < <s B ) } /\ E! y e. { x e. No | ( A < <s { x } /\ { x } < <s B ) } ( bday ` y ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) ) -> ( ( bday ` ( A |s B ) ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) <-> ( iota_ y e. { x e. No | ( A < <s { x } /\ { x } < <s B ) } ( bday ` y ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) ) = ( A |s B ) ) )
16	11, 12, 15	syl2anc 693	. 2 |- ( A < <s B -> ( ( bday ` ( A |s B ) ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) <-> ( iota_ y e. { x e. No | ( A < <s { x } /\ { x } < <s B ) } ( bday ` y ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) ) = ( A |s B ) ) )
17	2, 16	mpbird 247	1 |- ( A < <s B -> ( bday ` ( A |s B ) ) = |^| ( bday " { x e. No | ( A < <s { x } /\ { x } < <s B ) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E!wreu 2914   {crab 2916   {csn 4177   |^|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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798  df-sslt 31897  df-scut 31899

This theorem is referenced by:  scutun12  31917  scutbdaybnd  31921  scutbdaylt  31922