![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfunsn | Structured version Visualization version GIF version |
Description: If the restriction of a class to a singleton is not a function, then its value is the empty set. (An artifact of our function value definition.) (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 2567 | . . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦) | |
2 | vex 3466 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
3 | 2 | brresi 5998 | . . . . . . . . 9 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)) |
4 | velsn 4649 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
5 | breq1 5156 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
6 | 4, 5 | sylbi 216 | . . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) |
7 | 6 | biimpa 475 | . . . . . . . . 9 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) → 𝐴𝐹𝑦) |
8 | 3, 7 | sylbi 216 | . . . . . . . 8 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 → 𝐴𝐹𝑦) |
9 | 8 | moimi 2534 | . . . . . . 7 ⊢ (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) |
10 | 1, 9 | syl 17 | . . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) |
11 | tz6.12-2 6889 | . . . . . 6 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅) | |
12 | 10, 11 | nsyl4 158 | . . . . 5 ⊢ (¬ (𝐹‘𝐴) = ∅ → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) |
13 | 12 | alrimiv 1923 | . . . 4 ⊢ (¬ (𝐹‘𝐴) = ∅ → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) |
14 | relres 6015 | . . . 4 ⊢ Rel (𝐹 ↾ {𝐴}) | |
15 | 13, 14 | jctil 518 | . . 3 ⊢ (¬ (𝐹‘𝐴) = ∅ → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)) |
16 | dffun6 6567 | . . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)) | |
17 | 15, 16 | sylibr 233 | . 2 ⊢ (¬ (𝐹‘𝐴) = ∅ → Fun (𝐹 ↾ {𝐴})) |
18 | 17 | con1i 147 | 1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1532 = wceq 1534 ∈ wcel 2099 ∃*wmo 2527 ∃!weu 2557 ∅c0 4325 {csn 4633 class class class wbr 5153 ↾ cres 5684 Rel wrel 5687 Fun wfun 6548 ‘cfv 6554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-res 5694 df-iota 6506 df-fun 6556 df-fv 6562 |
This theorem is referenced by: fvfundmfvn0 6944 dffv2 6997 afv2ndeffv0 46873 |
Copyright terms: Public domain | W3C validator |