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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 5436 | . . . 4 ⊢ {𝑋} ∈ V | |
| 3 | 1, 2 | elmap 8911 | . . 3 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵) |
| 4 | mapsncnv.x | . . . . . 6 ⊢ 𝑋 ∈ V | |
| 5 | 4 | fsn2 7156 | . . . . 5 ⊢ (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹‘𝑋) ∈ 𝐵 ∧ 𝐹 = {〈𝑋, (𝐹‘𝑋)〉})) |
| 6 | 5 | simprbi 496 | . . . 4 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = {〈𝑋, (𝐹‘𝑋)〉}) |
| 7 | mapsncnv.s | . . . . . 6 ⊢ 𝑆 = {𝑋} | |
| 8 | 7 | xpeq1i 5711 | . . . . 5 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)}) |
| 9 | fvex 6919 | . . . . . 6 ⊢ (𝐹‘𝑋) ∈ V | |
| 10 | 4, 9 | xpsn 7161 | . . . . 5 ⊢ ({𝑋} × {(𝐹‘𝑋)}) = {〈𝑋, (𝐹‘𝑋)〉} |
| 11 | 8, 10 | eqtr2i 2766 | . . . 4 ⊢ {〈𝑋, (𝐹‘𝑋)〉} = (𝑆 × {(𝐹‘𝑋)}) |
| 12 | 6, 11 | eqtrdi 2793 | . . 3 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
| 13 | 3, 12 | sylbi 217 | . 2 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
| 14 | 7 | oveq2i 7442 | . 2 ⊢ (𝐵 ↑m 𝑆) = (𝐵 ↑m {𝑋}) |
| 15 | 13, 14 | eleq2s 2859 | 1 ⊢ (𝐹 ∈ (𝐵 ↑m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 |
| This theorem is referenced by: mapsncnv 8933 fvcoe1 22209 coe1mul2lem1 22270 coe1mul2 22272 0prjspnrel 42637 |
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