![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 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 | pm2.1 895 | . . 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 27149 | . . . . 5 ⊢ (𝐴 ∈ No → Fun 𝐴) | |
5 | norn 27151 | . . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o}) | |
6 | fvelrn 7078 | . . . . . . . 8 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴) | |
7 | ssel 3975 | . . . . . . . 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 452 | . . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) ∈ {1o, 2o})) |
10 | 1oex 8475 | . . . . . . 7 ⊢ 1o ∈ V | |
11 | 2on 8479 | . . . . . . . 8 ⊢ 2o ∈ On | |
12 | 11 | elexi 3493 | . . . . . . 7 ⊢ 2o ∈ V |
13 | 10, 12 | elpr2 4653 | . . . . . 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 584 | . . . 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 1090 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ⊆ wss 3948 ∅c0 4322 {cpr 4630 dom cdm 5676 ran crn 5677 Oncon0 6364 Fun wfun 6537 ‘cfv 6543 1oc1o 8458 2oc2o 8459 No csur 27140 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-1o 8465 df-2o 8466 df-no 27143 |
This theorem is referenced by: nolesgn2o 27171 nogesgn1o 27173 nosep1o 27181 nosep2o 27182 nolt02o 27195 nogt01o 27196 nosupbnd1lem5 27212 nosupbnd1lem6 27213 noinfbnd1lem5 27227 noinfbnd1lem6 27228 |
Copyright terms: Public domain | W3C validator |