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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 9069 | . . . 4 ⊢ {𝑋} ∈ Fin | |
2 | snopsuppss 8184 | . . . 4 ⊢ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋} | |
3 | 1, 2 | pm3.2i 469 | . . 3 ⊢ ({𝑋} ∈ Fin ∧ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋}) |
4 | ssfi 9198 | . . 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 6605 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → Fun {〈𝑋, 𝑌〉}) | |
7 | 6 | 3adant3 1129 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → Fun {〈𝑋, 𝑌〉}) |
8 | snex 5433 | . . . 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 9393 | . . 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 256 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 Vcvv 3461 ⊆ wss 3944 {csn 4630 〈cop 4636 class class class wbr 5149 Fun wfun 6543 (class class class)co 7419 supp csupp 8165 Fincfn 8964 finSupp cfsupp 9387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-supp 8166 df-1o 8487 df-en 8965 df-fin 8968 df-fsupp 9388 |
This theorem is referenced by: funsnfsupp 9417 |
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