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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 8307 | . . . 4 ⊢ {𝑋} ∈ Fin | |
2 | snopsuppss 7574 | . . . 4 ⊢ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋} | |
3 | 1, 2 | pm3.2i 464 | . . 3 ⊢ ({𝑋} ∈ Fin ∧ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋}) |
4 | ssfi 8449 | . . 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 6173 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → Fun {〈𝑋, 𝑌〉}) | |
7 | 6 | 3adant3 1168 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → Fun {〈𝑋, 𝑌〉}) |
8 | snex 5129 | . . . 4 ⊢ {〈𝑋, 𝑌〉} ∈ V | |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} ∈ V) |
10 | simp3 1174 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝑍 ∈ 𝑈) | |
11 | funisfsupp 8549 | . . 3 ⊢ ((Fun {〈𝑋, 𝑌〉} ∧ {〈𝑋, 𝑌〉} ∈ V ∧ 𝑍 ∈ 𝑈) → ({〈𝑋, 𝑌〉} finSupp 𝑍 ↔ ({〈𝑋, 𝑌〉} supp 𝑍) ∈ Fin)) | |
12 | 7, 9, 10, 11 | syl3anc 1496 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({〈𝑋, 𝑌〉} finSupp 𝑍 ↔ ({〈𝑋, 𝑌〉} supp 𝑍) ∈ Fin)) |
13 | 5, 12 | mpbird 249 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 ∈ wcel 2166 Vcvv 3414 ⊆ wss 3798 {csn 4397 〈cop 4403 class class class wbr 4873 Fun wfun 6117 (class class class)co 6905 supp csupp 7559 Fincfn 8222 finSupp cfsupp 8544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-supp 7560 df-1o 7826 df-er 8009 df-en 8223 df-fin 8226 df-fsupp 8545 |
This theorem is referenced by: funsnfsupp 8568 |
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