Theorem nofv 27509

Description: The function value of a surreal is either a sign or the empty set. (Contributed by Scott Fenton, 22-Jun-2011.)

Assertion

Ref	Expression
nofv	⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))

Proof of Theorem nofv

Step	Hyp	Ref	Expression
1		pm2.1 893	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴)
2		ndmfv 6917	. . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅)
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅))
4		nofun 27501	. . . . 5 ⊢ (𝐴 ∈ No → Fun 𝐴)
5		norn 27503	. . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
6		fvelrn 7069	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴)
7		ssel 3968	. . . . . . . 8 ⊢ (ran 𝐴 ⊆ {1o, 2o} → ((𝐴‘𝑋) ∈ ran 𝐴 → (𝐴‘𝑋) ∈ {1o, 2o}))
8	6, 7	syl5com 31	. . . . . . 7 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (ran 𝐴 ⊆ {1o, 2o} → (𝐴‘𝑋) ∈ {1o, 2o}))
9	8	impancom 451	. . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) ∈ {1o, 2o}))
10		1oex 8472	. . . . . . 7 ⊢ 1o ∈ V
11		2on 8476	. . . . . . . 8 ⊢ 2o ∈ On
12	11	elexi 3486	. . . . . . 7 ⊢ 2o ∈ V
13	10, 12	elpr2 4646	. . . . . 6 ⊢ ((𝐴‘𝑋) ∈ {1o, 2o} ↔ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))
14	9, 13	imbitrdi 250	. . . . 5 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)))
15	4, 5, 14	syl2anc 583	. . . 4 ⊢ (𝐴 ∈ No → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)))
16	3, 15	orim12d 961	. . 3 ⊢ (𝐴 ∈ No → ((¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴) → ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))))
17	1, 16	mpi 20	. 2 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)))
18		3orass 1087	. 2 ⊢ (((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o) ↔ ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)))
19	17, 18	sylibr 233	1 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 844   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098   ⊆ wss 3941  ∅c0 4315  {cpr 4623  dom cdm 5667  ran crn 5668  Oncon0 6355  Fun wfun 6528  ‘cfv 6534  1_oc1o 8455  2_oc2o 8456   No csur 27492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-1o 8462  df-2o 8463  df-no 27495

This theorem is referenced by:  nolesgn2o  27523  nogesgn1o  27525  nosep1o  27533  nosep2o  27534  nolt02o  27547  nogt01o  27548  nosupbnd1lem5  27564  nosupbnd1lem6  27565  noinfbnd1lem5  27579  noinfbnd1lem6  27580