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Mirrors > Home > MPE Home > Th. List > Mathboxes > funressnmo | 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 |
---|---|
funressnmo | ⊢ ((𝐴 ∈ 𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4551 | . . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | reseq2d 5851 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐴})) |
3 | 2 | funeqd 6402 | . . . 4 ⊢ (𝑥 = 𝐴 → (Fun (𝐹 ↾ {𝑥}) ↔ Fun (𝐹 ↾ {𝐴}))) |
4 | breq1 5056 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
5 | 4 | mobidv 2548 | . . . 4 ⊢ (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦)) |
6 | 3, 5 | imbi12d 348 | . . 3 ⊢ (𝑥 = 𝐴 → ((Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun (𝐹 ↾ {𝐴}) → ∃*𝑦 𝐴𝐹𝑦))) |
7 | funressnvmo 44211 | . . 3 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦) | |
8 | 6, 7 | vtoclg 3481 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Fun (𝐹 ↾ {𝐴}) → ∃*𝑦 𝐴𝐹𝑦)) |
9 | 8 | imp 410 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ Fun (𝐹 ↾ {𝐴})) → ∃*𝑦 𝐴𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃*wmo 2537 {csn 4541 class class class wbr 5053 ↾ cres 5553 Fun wfun 6374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-res 5563 df-fun 6382 |
This theorem is referenced by: funressneu 44213 |
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