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Mathbox for Alexander van der Vekens |
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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 7141 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅) → (𝐹:{𝑋}⟶{𝑌} ↔ 𝐹 = {〈𝑋, 𝑌〉})) | |
3 | 2 | 3adant3 1129 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → (𝐹:{𝑋}⟶{𝑌} ↔ 𝐹 = {〈𝑋, 𝑌〉})) |
4 | 1, 3 | mpbiri 257 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹:{𝑋}⟶{𝑌}) |
5 | snssi 4808 | . . . 4 ⊢ (𝑌 ∈ 𝑅 → {𝑌} ⊆ 𝑅) | |
6 | 5 | 3ad2ant2 1131 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → {𝑌} ⊆ 𝑅) |
7 | 4, 6 | fssd 6735 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹:{𝑋}⟶𝑅) |
8 | simp3 1135 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝑅 ∈ 𝑊) | |
9 | snex 5428 | . . 3 ⊢ {𝑋} ∈ V | |
10 | elmapg 8858 | . . 3 ⊢ ((𝑅 ∈ 𝑊 ∧ {𝑋} ∈ V) → (𝐹 ∈ (𝑅 ↑m {𝑋}) ↔ 𝐹:{𝑋}⟶𝑅)) | |
11 | 8, 9, 10 | sylancl 584 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → (𝐹 ∈ (𝑅 ↑m {𝑋}) ↔ 𝐹:{𝑋}⟶𝑅)) |
12 | 7, 11 | mpbird 256 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹 ∈ (𝑅 ↑m {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 Vcvv 3463 ⊆ wss 3947 {csn 4624 〈cop 4630 ⟶wf 6540 (class class class)co 7414 ↑m cmap 8845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8847 |
This theorem is referenced by: lincvalsng 47833 lcosn0 47837 |
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