![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > mapsnconst | 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 | . . . . 5 ⊢ 𝐵 ∈ V | |
2 | mapsncnv.x | . . . . . 6 ⊢ 𝑋 ∈ V | |
3 | 2 | snex 4117 | . . . . 5 ⊢ {𝑋} ∈ V |
4 | 1, 3 | elmap 6579 | . . . 4 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵) |
5 | 2 | fsn2 5602 | . . . . 5 ⊢ (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹‘𝑋) ∈ 𝐵 ∧ 𝐹 = {〈𝑋, (𝐹‘𝑋)〉})) |
6 | 5 | simprbi 273 | . . . 4 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = {〈𝑋, (𝐹‘𝑋)〉}) |
7 | 4, 6 | sylbi 120 | . . 3 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 {𝑋}) → 𝐹 = {〈𝑋, (𝐹‘𝑋)〉}) |
8 | mapsncnv.s | . . . 4 ⊢ 𝑆 = {𝑋} | |
9 | 8 | oveq2i 5793 | . . 3 ⊢ (𝐵 ↑𝑚 𝑆) = (𝐵 ↑𝑚 {𝑋}) |
10 | 7, 9 | eleq2s 2235 | . 2 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 𝑆) → 𝐹 = {〈𝑋, (𝐹‘𝑋)〉}) |
11 | 8 | xpeq1i 4567 | . . 3 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)}) |
12 | fvexg 5448 | . . . . 5 ⊢ ((𝐹 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑋 ∈ V) → (𝐹‘𝑋) ∈ V) | |
13 | 2, 12 | mpan2 422 | . . . 4 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 𝑆) → (𝐹‘𝑋) ∈ V) |
14 | xpsng 5603 | . . . 4 ⊢ ((𝑋 ∈ V ∧ (𝐹‘𝑋) ∈ V) → ({𝑋} × {(𝐹‘𝑋)}) = {〈𝑋, (𝐹‘𝑋)〉}) | |
15 | 2, 13, 14 | sylancr 411 | . . 3 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 𝑆) → ({𝑋} × {(𝐹‘𝑋)}) = {〈𝑋, (𝐹‘𝑋)〉}) |
16 | 11, 15 | syl5req 2186 | . 2 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 𝑆) → {〈𝑋, (𝐹‘𝑋)〉} = (𝑆 × {(𝐹‘𝑋)})) |
17 | 10, 16 | eqtrd 2173 | 1 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 Vcvv 2689 {csn 3532 〈cop 3535 × cxp 4545 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 ↑𝑚 cmap 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-map 6552 |
This theorem is referenced by: mapsncnv 6597 |
Copyright terms: Public domain | W3C validator |