Theorem nofv 27577

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		pm2.1 895	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴)
2		ndmfv 6926	. . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅)
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅))
4		nofun 27569	. . . . 5 ⊢ (𝐴 ∈ No → Fun 𝐴)
5		norn 27571	. . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
6		fvelrn 7080	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴)
7		ssel 3971	. . . . . . . 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 8490	. . . . . . 7 ⊢ 1o ∈ V
11		2on 8494	. . . . . . . 8 ⊢ 2o ∈ On
12	11	elexi 3489	. . . . . . 7 ⊢ 2o ∈ V
13	10, 12	elpr2 4649	. . . . . 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 963	. . 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 1088	. 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 846   ∨ w3o 1084   = wceq 1534   ∈ wcel 2099   ⊆ wss 3944  ∅c0 4318  {cpr 4626  dom cdm 5672  ran crn 5673  Oncon0 6363  Fun wfun 6536  ‘cfv 6542  1_oc1o 8473  2_oc2o 8474   No csur 27560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1o 8480  df-2o 8481  df-no 27563

This theorem is referenced by:  nolesgn2o  27591  nogesgn1o  27593  nosep1o  27601  nosep2o  27602  nolt02o  27615  nogt01o  27616  nosupbnd1lem5  27632  nosupbnd1lem6  27633  noinfbnd1lem5  27647  noinfbnd1lem6  27648