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Mirrors > Home > MPE Home > Th. List > snopfsupp | Structured version Visualization version GIF version |
Description: A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.) |
Ref | Expression |
---|---|
snopfsupp | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snfi 8577 | . . . 4 ⊢ {𝑋} ∈ Fin | |
2 | snopsuppss 7828 | . . . 4 ⊢ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋} | |
3 | 1, 2 | pm3.2i 474 | . . 3 ⊢ ({𝑋} ∈ Fin ∧ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋}) |
4 | ssfi 8722 | . . 3 ⊢ (({𝑋} ∈ Fin ∧ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋}) → ({〈𝑋, 𝑌〉} supp 𝑍) ∈ Fin) | |
5 | 3, 4 | mp1i 13 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({〈𝑋, 𝑌〉} supp 𝑍) ∈ Fin) |
6 | funsng 6375 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → Fun {〈𝑋, 𝑌〉}) | |
7 | 6 | 3adant3 1129 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → Fun {〈𝑋, 𝑌〉}) |
8 | snex 5297 | . . . 4 ⊢ {〈𝑋, 𝑌〉} ∈ V | |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} ∈ V) |
10 | simp3 1135 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝑍 ∈ 𝑈) | |
11 | funisfsupp 8822 | . . 3 ⊢ ((Fun {〈𝑋, 𝑌〉} ∧ {〈𝑋, 𝑌〉} ∈ V ∧ 𝑍 ∈ 𝑈) → ({〈𝑋, 𝑌〉} finSupp 𝑍 ↔ ({〈𝑋, 𝑌〉} supp 𝑍) ∈ Fin)) | |
12 | 7, 9, 10, 11 | syl3anc 1368 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({〈𝑋, 𝑌〉} finSupp 𝑍 ↔ ({〈𝑋, 𝑌〉} supp 𝑍) ∈ Fin)) |
13 | 5, 12 | mpbird 260 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 {csn 4525 〈cop 4531 class class class wbr 5030 Fun wfun 6318 (class class class)co 7135 supp csupp 7813 Fincfn 8492 finSupp cfsupp 8817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-supp 7814 df-1o 8085 df-er 8272 df-en 8493 df-fin 8496 df-fsupp 8818 |
This theorem is referenced by: funsnfsupp 8841 |
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