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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > funressnvmo | Structured version Visualization version GIF version |
Description: A function restricted to a singleton has at most one value for the singleton element as argument. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
funressnvmo | ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun6 6339 | . 2 ⊢ (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦)) | |
2 | breq1 5033 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑧(𝐹 ↾ {𝑥})𝑦)) | |
3 | 2 | equcoms 2027 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑧(𝐹 ↾ {𝑥})𝑦)) |
4 | 3 | biimpd 232 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦 → 𝑧(𝐹 ↾ {𝑥})𝑦)) |
5 | 4 | moimdv 2604 | . . . 4 ⊢ (𝑧 = 𝑥 → (∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦)) |
6 | 5 | spimvw 2002 | . . 3 ⊢ (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦) |
7 | vsnid 4562 | . . . . . 6 ⊢ 𝑥 ∈ {𝑥} | |
8 | vex 3444 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 8 | brresi 5827 | . . . . . 6 ⊢ (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ (𝑥 ∈ {𝑥} ∧ 𝑥𝐹𝑦)) |
10 | 7, 9 | mpbiran 708 | . . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑥𝐹𝑦) |
11 | 10 | biimpri 231 | . . . 4 ⊢ (𝑥𝐹𝑦 → 𝑥(𝐹 ↾ {𝑥})𝑦) |
12 | 11 | moimi 2603 | . . 3 ⊢ (∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦) |
13 | 6, 12 | syl 17 | . 2 ⊢ (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦) |
14 | 1, 13 | simplbiim 508 | 1 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 ∈ wcel 2111 ∃*wmo 2596 {csn 4525 class class class wbr 5030 ↾ cres 5521 Rel wrel 5524 Fun wfun 6318 |
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-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-cnv 5527 df-co 5528 df-res 5531 df-fun 6326 |
This theorem is referenced by: funressnmo 43638 funressndmafv2rn 43779 |
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