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 6449 | . 2 ⊢ (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦)) | |
2 | breq1 5077 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑧(𝐹 ↾ {𝑥})𝑦)) | |
3 | 2 | equcoms 2023 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑧(𝐹 ↾ {𝑥})𝑦)) |
4 | 3 | biimpd 228 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦 → 𝑧(𝐹 ↾ {𝑥})𝑦)) |
5 | 4 | moimdv 2546 | . . . 4 ⊢ (𝑧 = 𝑥 → (∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦)) |
6 | 5 | spimvw 1999 | . . 3 ⊢ (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦) |
7 | vsnid 4598 | . . . . . 6 ⊢ 𝑥 ∈ {𝑥} | |
8 | vex 3436 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 8 | brresi 5900 | . . . . . 6 ⊢ (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ (𝑥 ∈ {𝑥} ∧ 𝑥𝐹𝑦)) |
10 | 7, 9 | mpbiran 706 | . . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑥𝐹𝑦) |
11 | 10 | biimpri 227 | . . . 4 ⊢ (𝑥𝐹𝑦 → 𝑥(𝐹 ↾ {𝑥})𝑦) |
12 | 11 | moimi 2545 | . . 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 505 | 1 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 ∈ wcel 2106 ∃*wmo 2538 {csn 4561 class class class wbr 5074 ↾ cres 5591 Rel wrel 5594 Fun wfun 6427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-res 5601 df-fun 6435 |
This theorem is referenced by: funressnmo 44540 funressndmafv2rn 44715 |
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