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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapsnop | Structured version Visualization version GIF version |
Description: A singleton of an ordered pair as an element of the mapping operation. (Contributed by AV, 12-Apr-2019.) |
Ref | Expression |
---|---|
mapsnop.f | ⊢ 𝐹 = {〈𝑋, 𝑌〉} |
Ref | Expression |
---|---|
mapsnop | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹 ∈ (𝑅 ↑𝑚 {𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapsnop.f | . . . 4 ⊢ 𝐹 = {〈𝑋, 𝑌〉} | |
2 | fsng 6310 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅) → (𝐹:{𝑋}⟶{𝑌} ↔ 𝐹 = {〈𝑋, 𝑌〉})) | |
3 | 2 | 3adant3 1074 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → (𝐹:{𝑋}⟶{𝑌} ↔ 𝐹 = {〈𝑋, 𝑌〉})) |
4 | 1, 3 | mpbiri 247 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹:{𝑋}⟶{𝑌}) |
5 | snssi 4280 | . . . 4 ⊢ (𝑌 ∈ 𝑅 → {𝑌} ⊆ 𝑅) | |
6 | 5 | 3ad2ant2 1076 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → {𝑌} ⊆ 𝑅) |
7 | 4, 6 | fssd 5970 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹:{𝑋}⟶𝑅) |
8 | simp3 1056 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝑅 ∈ 𝑊) | |
9 | snex 4835 | . . 3 ⊢ {𝑋} ∈ V | |
10 | elmapg 7757 | . . 3 ⊢ ((𝑅 ∈ 𝑊 ∧ {𝑋} ∈ V) → (𝐹 ∈ (𝑅 ↑𝑚 {𝑋}) ↔ 𝐹:{𝑋}⟶𝑅)) | |
11 | 8, 9, 10 | sylancl 693 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → (𝐹 ∈ (𝑅 ↑𝑚 {𝑋}) ↔ 𝐹:{𝑋}⟶𝑅)) |
12 | 7, 11 | mpbird 246 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹 ∈ (𝑅 ↑𝑚 {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {csn 4125 〈cop 4131 ⟶wf 5800 (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-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-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 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: lincvalsng 41999 lcosn0 42003 |
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