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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 8596 | . . . 4 ⊢ {𝑋} ∈ Fin | |
2 | snopsuppss 7847 | . . . 4 ⊢ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋} | |
3 | 1, 2 | pm3.2i 473 | . . 3 ⊢ ({𝑋} ∈ Fin ∧ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋}) |
4 | ssfi 8740 | . . 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 6407 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → Fun {〈𝑋, 𝑌〉}) | |
7 | 6 | 3adant3 1128 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → Fun {〈𝑋, 𝑌〉}) |
8 | snex 5334 | . . . 4 ⊢ {〈𝑋, 𝑌〉} ∈ V | |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} ∈ V) |
10 | simp3 1134 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝑍 ∈ 𝑈) | |
11 | funisfsupp 8840 | . . 3 ⊢ ((Fun {〈𝑋, 𝑌〉} ∧ {〈𝑋, 𝑌〉} ∈ V ∧ 𝑍 ∈ 𝑈) → ({〈𝑋, 𝑌〉} finSupp 𝑍 ↔ ({〈𝑋, 𝑌〉} supp 𝑍) ∈ Fin)) | |
12 | 7, 9, 10, 11 | syl3anc 1367 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({〈𝑋, 𝑌〉} finSupp 𝑍 ↔ ({〈𝑋, 𝑌〉} supp 𝑍) ∈ Fin)) |
13 | 5, 12 | mpbird 259 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 {csn 4569 〈cop 4575 class class class wbr 5068 Fun wfun 6351 (class class class)co 7158 supp csupp 7832 Fincfn 8511 finSupp cfsupp 8835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-supp 7833 df-1o 8104 df-er 8291 df-en 8512 df-fin 8515 df-fsupp 8836 |
This theorem is referenced by: funsnfsupp 8859 |
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