Theorem nofv 31810

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		pm2.1 433	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴)
2		ndmfv 6218	. . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅)
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅))
4		nofun 31802	. . . . 5 ⊢ (𝐴 ∈ No → Fun 𝐴)
5		norn 31804	. . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
6		fvelrn 6352	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴)
7		ssel 3597	. . . . . . . 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 456	. . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) ∈ {1𝑜, 2𝑜}))
10		1on 7567	. . . . . . . 8 ⊢ 1𝑜 ∈ On
11	10	elexi 3213	. . . . . . 7 ⊢ 1𝑜 ∈ V
12		2on 7568	. . . . . . . 8 ⊢ 2𝑜 ∈ On
13	12	elexi 3213	. . . . . . 7 ⊢ 2𝑜 ∈ V
14	11, 13	elpr2 4199	. . . . . 6 ⊢ ((𝐴‘𝑋) ∈ {1𝑜, 2𝑜} ↔ ((𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜))
15	9, 14	syl6ib 241	. . . . 5 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜)))
16	4, 5, 15	syl2anc 693	. . . 4 ⊢ (𝐴 ∈ No → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜)))
17	3, 16	orim12d 883	. . 3 ⊢ (𝐴 ∈ No → ((¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴) → ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜))))
18	1, 17	mpi 20	. 2 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜)))
19		3orass 1040	. 2 ⊢ (((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜) ↔ ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜)))
20	18, 19	sylibr 224	1 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ∅c0 3915  {cpr 4179  dom cdm 5114  ran crn 5115  Oncon0 5723  Fun wfun 5882  ‘cfv 5888  1_𝑜c1o 7553  2_𝑜c2o 7554   No csur 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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  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-eprel 5029  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-suc 5729  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-1o 7560  df-2o 7561  df-no 31796

This theorem is referenced by:  nolesgn2o  31824  nosep1o  31832  nolt02o  31845  nosupbnd1lem5  31858  nosupbnd1lem6  31859