Theorem noreson 27904

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 27890	. . 3 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
2		onin 6393	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∩ 𝐵) ∈ On)
3		fresin 6748	. . . . . . . 8 ⊢ (𝐴:𝑥⟶{1o, 2o} → (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o})
4		feq2 6685	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝐴 ↾ 𝐵):𝑦⟶{1o, 2o} ↔ (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o}))
5	4	rspcev 3579	. . . . . . . 8 ⊢ (((𝑥 ∩ 𝐵) ∈ On ∧ (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
6	2, 3, 5	syl2an 608	. . . . . . 7 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴:𝑥⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
7	6	an32s 665	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
8	7	ex 418	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}))
9	8	rexlimiva 3157	. . . 4 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}))
10	9	imp 412	. . 3 ⊢ ((∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
11	1, 10	sylanb 593	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
12		elno 27890	. 2 ⊢ ((𝐴 ↾ 𝐵) ∈ No ↔ ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
13	11, 12	sylibr 237	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ↾ 𝐵) ∈ No )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∃wrex 3088   ∩ cin 3901  {cpr 4589   ↾ cres 5661  Oncon0 6361  ⟶wf 6533  1_oc1o 8452  2_oc2o 8453   No csur 27884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ord 6364  df-on 6365  df-fun 6539  df-fn 6540  df-f 6541  df-no 27887

This theorem is used by:  ltsres  27906  nodenselem6  27933  noresle  27941  nosupbnd1lem1  27952  nosupbnd1lem2  27953  nosupbnd1lem6  27957  nosupbnd1  27958  nosupbnd2lem1  27959  nosupbnd2  27960  noinfbnd1lem1  27967  noinfbnd1lem2  27968  noinfbnd1lem6  27972  noinfbnd1  27973  noinfbnd2lem1  27974  noinfbnd2  27975  nosupinfsep  27976  noetasuplem4  27980  noetainflem4  27984