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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 6631 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅) → (𝐹:{𝑋}⟶{𝑌} ↔ 𝐹 = {〈𝑋, 𝑌〉})) | |
3 | 2 | 3adant3 1163 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → (𝐹:{𝑋}⟶{𝑌} ↔ 𝐹 = {〈𝑋, 𝑌〉})) |
4 | 1, 3 | mpbiri 250 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹:{𝑋}⟶{𝑌}) |
5 | snssi 4527 | . . . 4 ⊢ (𝑌 ∈ 𝑅 → {𝑌} ⊆ 𝑅) | |
6 | 5 | 3ad2ant2 1165 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → {𝑌} ⊆ 𝑅) |
7 | 4, 6 | fssd 6270 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹:{𝑋}⟶𝑅) |
8 | simp3 1169 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝑅 ∈ 𝑊) | |
9 | snex 5099 | . . 3 ⊢ {𝑋} ∈ V | |
10 | elmapg 8108 | . . 3 ⊢ ((𝑅 ∈ 𝑊 ∧ {𝑋} ∈ V) → (𝐹 ∈ (𝑅 ↑𝑚 {𝑋}) ↔ 𝐹:{𝑋}⟶𝑅)) | |
11 | 8, 9, 10 | sylancl 581 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → (𝐹 ∈ (𝑅 ↑𝑚 {𝑋}) ↔ 𝐹:{𝑋}⟶𝑅)) |
12 | 7, 11 | mpbird 249 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹 ∈ (𝑅 ↑𝑚 {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ⊆ wss 3769 {csn 4368 〈cop 4374 ⟶wf 6097 (class class class)co 6878 ↑𝑚 cmap 8095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-map 8097 |
This theorem is referenced by: lincvalsng 43004 lcosn0 43008 |
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