Theorem noreson 33549

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 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ↾ 𝐵) ∈ No )

Proof of Theorem noreson

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

Step	Hyp	Ref	Expression
1		elno 33535	. . 3 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
2		onin 6222	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∩ 𝐵) ∈ On)
3		fresin 6566	. . . . . . . 8 ⊢ (𝐴:𝑥⟶{1o, 2o} → (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o})
4		feq2 6505	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝐴 ↾ 𝐵):𝑦⟶{1o, 2o} ↔ (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o}))
5	4	rspcev 3527	. . . . . . . 8 ⊢ (((𝑥 ∩ 𝐵) ∈ On ∧ (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
6	2, 3, 5	syl2an 599	. . . . . . 7 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴:𝑥⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
7	6	an32s 652	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
8	7	ex 416	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}))
9	8	rexlimiva 3190	. . . 4 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}))
10	9	imp 410	. . 3 ⊢ ((∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
11	1, 10	sylanb 584	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
12		elno 33535	. 2 ⊢ ((𝐴 ↾ 𝐵) ∈ No ↔ ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
13	11, 12	sylibr 237	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ↾ 𝐵) ∈ No )

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2112  ∃wrex 3052   ∩ cin 3852  {cpr 4529   ↾ cres 5538  Oncon0 6191  ⟶wf 6354  1_oc1o 8173  2_oc2o 8174   No csur 33529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6194  df-on 6195  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-no 33532

This theorem is referenced by:  sltres  33551  nodenselem6  33578  noresle  33586  nosupbnd1lem1  33597  nosupbnd1lem2  33598  nosupbnd1lem6  33602  nosupbnd1  33603  nosupbnd2lem1  33604  nosupbnd2  33605  noinfbnd1lem1  33612  noinfbnd1lem2  33613  noinfbnd1lem6  33617  noinfbnd1  33618  noinfbnd2lem1  33619  noinfbnd2  33620  nosupinfsep  33621  noetasuplem4  33625  noetainflem4  33629