![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nfunsn | GIF version |
Description: If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nfunsn | ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 1980 | . . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦) | |
2 | vex 2622 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
3 | 2 | brres 4719 | . . . . . . . . 9 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴})) |
4 | velsn 3463 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
5 | breq1 3848 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
6 | 4, 5 | sylbi 119 | . . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) |
7 | 6 | biimpac 292 | . . . . . . . . 9 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}) → 𝐴𝐹𝑦) |
8 | 3, 7 | sylbi 119 | . . . . . . . 8 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 → 𝐴𝐹𝑦) |
9 | 8 | moimi 2013 | . . . . . . 7 ⊢ (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) |
10 | 1, 9 | syl 14 | . . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) |
11 | 10 | alrimiv 1802 | . . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) |
12 | relres 4741 | . . . . 5 ⊢ Rel (𝐹 ↾ {𝐴}) | |
13 | 11, 12 | jctil 305 | . . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)) |
14 | dffun6 5029 | . . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)) | |
15 | 13, 14 | sylibr 132 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → Fun (𝐹 ↾ {𝐴})) |
16 | 15 | con3i 597 | . 2 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → ¬ ∃!𝑦 𝐴𝐹𝑦) |
17 | tz6.12-2 5296 | . 2 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅) | |
18 | 16, 17 | syl 14 | 1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1287 = wceq 1289 ∈ wcel 1438 ∃!weu 1948 ∃*wmo 1949 ∅c0 3286 {csn 3446 class class class wbr 3845 ↾ cres 4440 Rel wrel 4443 Fun wfun 5009 ‘cfv 5015 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-res 4450 df-iota 4980 df-fun 5017 df-fv 5023 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |