Theorem nofv 31054

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		pm2.1 432	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴)
2		ndmfv 6128	. . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅)
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅))
4		nofun 31046	. . . . 5 ⊢ (𝐴 ∈ No → Fun 𝐴)
5		norn 31048	. . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
6		fvelrn 6260	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴)
7		ssel 3562	. . . . . . . 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 455	. . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) ∈ {1𝑜, 2𝑜}))
10		1on 7454	. . . . . . . 8 ⊢ 1𝑜 ∈ On
11	10	elexi 3186	. . . . . . 7 ⊢ 1𝑜 ∈ V
12		2on 7455	. . . . . . . 8 ⊢ 2𝑜 ∈ On
13	12	elexi 3186	. . . . . . 7 ⊢ 2𝑜 ∈ V
14	11, 13	elpr2 4147	. . . . . 6 ⊢ ((𝐴‘𝑋) ∈ {1𝑜, 2𝑜} ↔ ((𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜))
15	9, 14	syl6ib 240	. . . . 5 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜)))
16	4, 5, 15	syl2anc 691	. . . 4 ⊢ (𝐴 ∈ No → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜)))
17	3, 16	orim12d 879	. . 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 1034	. 2 ⊢ (((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜) ↔ ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜)))
20	18, 19	sylibr 223	1 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∅c0 3874  {cpr 4127  dom cdm 5038  ran crn 5039  Oncon0 5640  Fun wfun 5798  ‘cfv 5804  1_𝑜c1o 7440  2_𝑜c2o 7441   No csur 31037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-2o 7448  df-no 31040

This theorem is referenced by:  nobndup  31099  nobnddown  31100