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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 | pm2.1 894 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴) | |
2 | ndmfv 6675 | . . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅)) |
4 | nofun 33269 | . . . . 5 ⊢ (𝐴 ∈ No → Fun 𝐴) | |
5 | norn 33271 | . . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o}) | |
6 | fvelrn 6821 | . . . . . . . 8 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴) | |
7 | ssel 3908 | . . . . . . . 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 | 1oex 8093 | . . . . . . 7 ⊢ 1o ∈ V | |
11 | 2on 8094 | . . . . . . . 8 ⊢ 2o ∈ On | |
12 | 11 | elexi 3460 | . . . . . . 7 ⊢ 2o ∈ V |
13 | 10, 12 | elpr2 4550 | . . . . . 6 ⊢ ((𝐴‘𝑋) ∈ {1o, 2o} ↔ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)) |
14 | 9, 13 | syl6ib 254 | . . . . 5 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))) |
15 | 4, 5, 14 | syl2anc 587 | . . . 4 ⊢ (𝐴 ∈ No → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))) |
16 | 3, 15 | orim12d 962 | . . 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 237 | 1 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 {cpr 4527 dom cdm 5519 ran crn 5520 Oncon0 6159 Fun wfun 6318 ‘cfv 6324 1oc1o 8078 2oc2o 8079 No csur 33260 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-1o 8085 df-2o 8086 df-no 33263 |
This theorem is referenced by: nolesgn2o 33291 nosep1o 33299 nolt02o 33312 nosupbnd1lem5 33325 nosupbnd1lem6 33326 |
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