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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, 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 2487 | . . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦) | |
2 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
3 | 2 | brres 5323 | . . . . . . . . 9 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴})) |
4 | velsn 4141 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
5 | breq1 4586 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
6 | 4, 5 | sylbi 206 | . . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) |
7 | 6 | biimpac 502 | . . . . . . . . 9 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}) → 𝐴𝐹𝑦) |
8 | 3, 7 | sylbi 206 | . . . . . . . 8 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 → 𝐴𝐹𝑦) |
9 | 8 | moimi 2508 | . . . . . . 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 6094 | . . . . . 6 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅) | |
12 | 10, 11 | nsyl4 155 | . . . . 5 ⊢ (¬ (𝐹‘𝐴) = ∅ → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) |
13 | 12 | alrimiv 1842 | . . . 4 ⊢ (¬ (𝐹‘𝐴) = ∅ → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) |
14 | relres 5346 | . . . 4 ⊢ Rel (𝐹 ↾ {𝐴}) | |
15 | 13, 14 | jctil 558 | . . 3 ⊢ (¬ (𝐹‘𝐴) = ∅ → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)) |
16 | dffun6 5819 | . . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)) | |
17 | 15, 16 | sylibr 223 | . 2 ⊢ (¬ (𝐹‘𝐴) = ∅ → Fun (𝐹 ↾ {𝐴})) |
18 | 17 | con1i 143 | 1 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∃!weu 2458 ∃*wmo 2459 ∅c0 3874 {csn 4125 class class class wbr 4583 ↾ cres 5040 Rel wrel 5043 Fun wfun 5798 ‘cfv 5804 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: fvfundmfvn0 6136 dffv2 6181 |
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