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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 5297 | . . . 4 ⊢ {𝑋} ∈ V | |
3 | 1, 2 | elmap 8418 | . . 3 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵) |
4 | mapsncnv.x | . . . . . 6 ⊢ 𝑋 ∈ V | |
5 | 4 | fsn2 6875 | . . . . 5 ⊢ (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹‘𝑋) ∈ 𝐵 ∧ 𝐹 = {〈𝑋, (𝐹‘𝑋)〉})) |
6 | 5 | simprbi 500 | . . . 4 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = {〈𝑋, (𝐹‘𝑋)〉}) |
7 | mapsncnv.s | . . . . . 6 ⊢ 𝑆 = {𝑋} | |
8 | 7 | xpeq1i 5545 | . . . . 5 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)}) |
9 | fvex 6658 | . . . . . 6 ⊢ (𝐹‘𝑋) ∈ V | |
10 | 4, 9 | xpsn 6880 | . . . . 5 ⊢ ({𝑋} × {(𝐹‘𝑋)}) = {〈𝑋, (𝐹‘𝑋)〉} |
11 | 8, 10 | eqtr2i 2822 | . . . 4 ⊢ {〈𝑋, (𝐹‘𝑋)〉} = (𝑆 × {(𝐹‘𝑋)}) |
12 | 6, 11 | eqtrdi 2849 | . . 3 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
13 | 3, 12 | sylbi 220 | . 2 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
14 | 7 | oveq2i 7146 | . 2 ⊢ (𝐵 ↑m 𝑆) = (𝐵 ↑m {𝑋}) |
15 | 13, 14 | eleq2s 2908 | 1 ⊢ (𝐹 ∈ (𝐵 ↑m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 〈cop 4531 × cxp 5517 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 |
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-pow 5231 ax-pr 5295 ax-un 7441 |
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-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 |
This theorem is referenced by: mapsncnv 8440 fvcoe1 20836 coe1mul2lem1 20896 coe1mul2 20898 0prjspnrel 39613 |
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