![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mapsnconst | Structured version Visualization version GIF version |
Description: Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
Ref | Expression |
---|---|
mapsncnv.s | ⊢ 𝑆 = {𝑋} |
mapsncnv.b | ⊢ 𝐵 ∈ V |
mapsncnv.x | ⊢ 𝑋 ∈ V |
Ref | Expression |
---|---|
mapsnconst | ⊢ (𝐹 ∈ (𝐵 ↑m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapsncnv.b | . . . 4 ⊢ 𝐵 ∈ V | |
2 | snex 5433 | . . . 4 ⊢ {𝑋} ∈ V | |
3 | 1, 2 | elmap 8890 | . . 3 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵) |
4 | mapsncnv.x | . . . . . 6 ⊢ 𝑋 ∈ V | |
5 | 4 | fsn2 7145 | . . . . 5 ⊢ (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹‘𝑋) ∈ 𝐵 ∧ 𝐹 = {〈𝑋, (𝐹‘𝑋)〉})) |
6 | 5 | simprbi 495 | . . . 4 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = {〈𝑋, (𝐹‘𝑋)〉}) |
7 | mapsncnv.s | . . . . . 6 ⊢ 𝑆 = {𝑋} | |
8 | 7 | xpeq1i 5704 | . . . . 5 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)}) |
9 | fvex 6909 | . . . . . 6 ⊢ (𝐹‘𝑋) ∈ V | |
10 | 4, 9 | xpsn 7150 | . . . . 5 ⊢ ({𝑋} × {(𝐹‘𝑋)}) = {〈𝑋, (𝐹‘𝑋)〉} |
11 | 8, 10 | eqtr2i 2754 | . . . 4 ⊢ {〈𝑋, (𝐹‘𝑋)〉} = (𝑆 × {(𝐹‘𝑋)}) |
12 | 6, 11 | eqtrdi 2781 | . . 3 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
13 | 3, 12 | sylbi 216 | . 2 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
14 | 7 | oveq2i 7430 | . 2 ⊢ (𝐵 ↑m 𝑆) = (𝐵 ↑m {𝑋}) |
15 | 13, 14 | eleq2s 2843 | 1 ⊢ (𝐹 ∈ (𝐵 ↑m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 {csn 4630 〈cop 4636 × cxp 5676 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-map 8847 |
This theorem is referenced by: mapsncnv 8912 fvcoe1 22150 coe1mul2lem1 22211 coe1mul2 22213 0prjspnrel 42186 |
Copyright terms: Public domain | W3C validator |