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Mirrors > Home > MPE Home > Th. List > Mathboxes > nofv | Structured version Visualization version GIF version |
Description: The function value of a surreal is either a sign or the empty set. (Contributed by Scott Fenton, 22-Jun-2011.) |
Ref | Expression |
---|---|
nofv | ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1𝑜 ∨ (𝐴‘𝑋) = 2𝑜)) |
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)) |
Colors of variables: wff setvar class |
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 1𝑜c1o 7440 2𝑜c2o 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 |
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