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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 7923 | . . . 4 ⊢ {𝑋} ∈ Fin | |
2 | snopsuppss 7197 | . . . 4 ⊢ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋} | |
3 | 1, 2 | pm3.2i 470 | . . 3 ⊢ ({𝑋} ∈ Fin ∧ ({〈𝑋, 𝑌〉} supp 𝑍) ⊆ {𝑋}) |
4 | ssfi 8065 | . . 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 5851 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → Fun {〈𝑋, 𝑌〉}) | |
7 | 6 | 3adant3 1074 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → Fun {〈𝑋, 𝑌〉}) |
8 | snex 4835 | . . . 4 ⊢ {〈𝑋, 𝑌〉} ∈ V | |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} ∈ V) |
10 | simp3 1056 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → 𝑍 ∈ 𝑈) | |
11 | funisfsupp 8163 | . . 3 ⊢ ((Fun {〈𝑋, 𝑌〉} ∧ {〈𝑋, 𝑌〉} ∈ V ∧ 𝑍 ∈ 𝑈) → ({〈𝑋, 𝑌〉} finSupp 𝑍 ↔ ({〈𝑋, 𝑌〉} supp 𝑍) ∈ Fin)) | |
12 | 7, 9, 10, 11 | syl3anc 1318 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → ({〈𝑋, 𝑌〉} finSupp 𝑍 ↔ ({〈𝑋, 𝑌〉} supp 𝑍) ∈ Fin)) |
13 | 5, 12 | mpbird 246 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → {〈𝑋, 𝑌〉} finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {csn 4125 〈cop 4131 class class class wbr 4583 Fun wfun 5798 (class class class)co 6549 supp csupp 7182 Fincfn 7841 finSupp cfsupp 8158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-supp 7183 df-1o 7447 df-er 7629 df-en 7842 df-fin 7845 df-fsupp 8159 |
This theorem is referenced by: funsnfsupp 8182 |
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