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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 5389 | . . . 4 ⊢ {𝑋} ∈ V | |
3 | 1, 2 | elmap 8812 | . . 3 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵) |
4 | mapsncnv.x | . . . . . 6 ⊢ 𝑋 ∈ V | |
5 | 4 | fsn2 7083 | . . . . 5 ⊢ (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹‘𝑋) ∈ 𝐵 ∧ 𝐹 = {⟨𝑋, (𝐹‘𝑋)⟩})) |
6 | 5 | simprbi 498 | . . . 4 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = {⟨𝑋, (𝐹‘𝑋)⟩}) |
7 | mapsncnv.s | . . . . . 6 ⊢ 𝑆 = {𝑋} | |
8 | 7 | xpeq1i 5660 | . . . . 5 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)}) |
9 | fvex 6856 | . . . . . 6 ⊢ (𝐹‘𝑋) ∈ V | |
10 | 4, 9 | xpsn 7088 | . . . . 5 ⊢ ({𝑋} × {(𝐹‘𝑋)}) = {⟨𝑋, (𝐹‘𝑋)⟩} |
11 | 8, 10 | eqtr2i 2762 | . . . 4 ⊢ {⟨𝑋, (𝐹‘𝑋)⟩} = (𝑆 × {(𝐹‘𝑋)}) |
12 | 6, 11 | eqtrdi 2789 | . . 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 7369 | . 2 ⊢ (𝐵 ↑m 𝑆) = (𝐵 ↑m {𝑋}) |
15 | 13, 14 | eleq2s 2852 | 1 ⊢ (𝐹 ∈ (𝐵 ↑m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 {csn 4587 ⟨cop 4593 × cxp 5632 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 ↑m cmap 8768 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8770 |
This theorem is referenced by: mapsncnv 8834 fvcoe1 21594 coe1mul2lem1 21654 coe1mul2 21656 0prjspnrel 41008 |
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