Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 6370 | . 2 ⊢ (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦)) | |
2 | breq1 5069 | . . . . . . 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 231 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦 → 𝑧(𝐹 ↾ {𝑥})𝑦)) |
5 | 4 | moimdv 2629 | . . . 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 4602 | . . . . . 6 ⊢ 𝑥 ∈ {𝑥} | |
8 | vex 3497 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 8 | brresi 5862 | . . . . . 6 ⊢ (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ (𝑥 ∈ {𝑥} ∧ 𝑥𝐹𝑦)) |
10 | 7, 9 | mpbiran 707 | . . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑥𝐹𝑦) |
11 | 10 | biimpri 230 | . . . 4 ⊢ (𝑥𝐹𝑦 → 𝑥(𝐹 ↾ {𝑥})𝑦) |
12 | 11 | moimi 2627 | . . 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 507 | 1 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∈ wcel 2114 ∃*wmo 2620 {csn 4567 class class class wbr 5066 ↾ cres 5557 Rel wrel 5560 Fun wfun 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-cnv 5563 df-co 5564 df-res 5567 df-fun 6357 |
This theorem is referenced by: funressnmo 43330 funressndmafv2rn 43471 |
Copyright terms: Public domain | W3C validator |