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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 4835 | . . . 4 ⊢ {𝑋} ∈ V | |
3 | 1, 2 | elmap 7772 | . . 3 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵) |
4 | mapsncnv.x | . . . . . 6 ⊢ 𝑋 ∈ V | |
5 | 4 | fsn2 6309 | . . . . 5 ⊢ (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹‘𝑋) ∈ 𝐵 ∧ 𝐹 = {〈𝑋, (𝐹‘𝑋)〉})) |
6 | 5 | simprbi 479 | . . . 4 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = {〈𝑋, (𝐹‘𝑋)〉}) |
7 | mapsncnv.s | . . . . . 6 ⊢ 𝑆 = {𝑋} | |
8 | 7 | xpeq1i 5059 | . . . . 5 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)}) |
9 | fvex 6113 | . . . . . 6 ⊢ (𝐹‘𝑋) ∈ V | |
10 | 4, 9 | xpsn 6313 | . . . . 5 ⊢ ({𝑋} × {(𝐹‘𝑋)}) = {〈𝑋, (𝐹‘𝑋)〉} |
11 | 8, 10 | eqtr2i 2633 | . . . 4 ⊢ {〈𝑋, (𝐹‘𝑋)〉} = (𝑆 × {(𝐹‘𝑋)}) |
12 | 6, 11 | syl6eq 2660 | . . 3 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
13 | 3, 12 | sylbi 206 | . 2 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 {𝑋}) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
14 | 7 | oveq2i 6560 | . 2 ⊢ (𝐵 ↑𝑚 𝑆) = (𝐵 ↑𝑚 {𝑋}) |
15 | 13, 14 | eleq2s 2706 | 1 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 |
This theorem is referenced by: mapsncnv 7790 fvcoe1 19398 coe1mul2lem1 19458 coe1mul2 19460 |
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