Theorem noreson 31813

Description: The restriction of a surreal to an ordinal is still a surreal. (Contributed by Scott Fenton, 4-Sep-2011.)

Assertion

Ref	Expression
noreson	|- ( ( A e. No /\ B e. On ) -> ( A |` B ) e. No )

Proof of Theorem noreson

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

Step	Hyp	Ref	Expression
1		elno 31799	. . 3 |- ( A e. No <-> E. x e. On A : x --> { 1o , 2o } )
2		onin 5754	. . . . . . . 8 |- ( ( x e. On /\ B e. On ) -> ( x i^i B ) e. On )
3		fresin 6073	. . . . . . . 8 |- ( A : x --> { 1o , 2o } -> ( A |` B ) : ( x i^i B ) --> { 1o , 2o } )
4		feq2 6027	. . . . . . . . 9 |- ( y = ( x i^i B ) -> ( ( A |` B ) : y --> { 1o , 2o } <-> ( A |` B ) : ( x i^i B ) --> { 1o , 2o } ) )
5	4	rspcev 3309	. . . . . . . 8 |- ( ( ( x i^i B ) e. On /\ ( A |` B ) : ( x i^i B ) --> { 1o , 2o } ) -> E. y e. On ( A |` B ) : y --> { 1o , 2o } )
6	2, 3, 5	syl2an 494	. . . . . . 7 |- ( ( ( x e. On /\ B e. On ) /\ A : x --> { 1o , 2o } ) -> E. y e. On ( A |` B ) : y --> { 1o , 2o } )
7	6	an32s 846	. . . . . 6 |- ( ( ( x e. On /\ A : x --> { 1o , 2o } ) /\ B e. On ) -> E. y e. On ( A |` B ) : y --> { 1o , 2o } )
8	7	ex 450	. . . . 5 |- ( ( x e. On /\ A : x --> { 1o , 2o } ) -> ( B e. On -> E. y e. On ( A |` B ) : y --> { 1o , 2o } ) )
9	8	rexlimiva 3028	. . . 4 |- ( E. x e. On A : x --> { 1o , 2o } -> ( B e. On -> E. y e. On ( A |` B ) : y --> { 1o , 2o } ) )
10	9	imp 445	. . 3 |- ( ( E. x e. On A : x --> { 1o , 2o } /\ B e. On ) -> E. y e. On ( A |` B ) : y --> { 1o , 2o } )
11	1, 10	sylanb 489	. 2 |- ( ( A e. No /\ B e. On ) -> E. y e. On ( A |` B ) : y --> { 1o , 2o } )
12		elno 31799	. 2 |- ( ( A |` B ) e. No <-> E. y e. On ( A |` B ) : y --> { 1o , 2o } )
13	11, 12	sylibr 224	1 |- ( ( A e. No /\ B e. On ) -> ( A |` B ) e. No )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   E.wrex 2913   i^i cin 3573   {cpr 4179   |` cres 5116   Oncon0 5723   -->wf 5884   1oc1o 7553   2oc2o 7554   Nocsur 31793

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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-no 31796

This theorem is referenced by:  sltres  31815  nodenselem6  31839  noresle  31846  nosupbnd1lem1  31854  nosupbnd1lem2  31855  nosupbnd1lem6  31859  nosupbnd1  31860  nosupbnd2lem1  31861  nosupbnd2  31862  noetalem3  31865