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Mirrors > Home > MPE Home > Th. List > fsuppsssupp | Structured version Visualization version GIF version |
Description: If the support of a function is a subset of the support of a finitely supported function, the function is finitely supported. (Contributed by AV, 2-Jul-2019.) (Proof shortened by AV, 15-Jul-2019.) |
Ref | Expression |
---|---|
fsuppsssupp | ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐹 finSupp 𝑍 ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) → 𝐺 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 766 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐹 finSupp 𝑍 ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) → 𝐺 ∈ 𝑉) | |
2 | simplr 768 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐹 finSupp 𝑍 ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) → Fun 𝐺) | |
3 | relfsupp 9388 | . . . 4 ⊢ Rel finSupp | |
4 | 3 | brrelex2i 5735 | . . 3 ⊢ (𝐹 finSupp 𝑍 → 𝑍 ∈ V) |
5 | 4 | ad2antrl 727 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐹 finSupp 𝑍 ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) → 𝑍 ∈ V) |
6 | id 22 | . . . . 5 ⊢ (𝐹 finSupp 𝑍 → 𝐹 finSupp 𝑍) | |
7 | 6 | fsuppimpd 9394 | . . . 4 ⊢ (𝐹 finSupp 𝑍 → (𝐹 supp 𝑍) ∈ Fin) |
8 | 7 | anim1i 614 | . . 3 ⊢ ((𝐹 finSupp 𝑍 ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐹 supp 𝑍) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) |
9 | 8 | adantl 481 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐹 finSupp 𝑍 ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) → ((𝐹 supp 𝑍) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) |
10 | suppssfifsupp 9404 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ V) ∧ ((𝐹 supp 𝑍) ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) → 𝐺 finSupp 𝑍) | |
11 | 1, 2, 5, 9, 10 | syl31anc 1371 | 1 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺) ∧ (𝐹 finSupp 𝑍 ∧ (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))) → 𝐺 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 Vcvv 3471 ⊆ wss 3947 class class class wbr 5148 Fun wfun 6542 (class class class)co 7420 supp csupp 8165 Fincfn 8964 finSupp cfsupp 9386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 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 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-1o 8487 df-en 8965 df-fin 8968 df-fsupp 9387 |
This theorem is referenced by: cantnflem1 9713 dprdfinv 19976 dmdprdsplitlem 19994 dpjidcl 20015 frlmphllem 21714 frlmphl 21715 rrxcph 25333 tdeglem4 26008 tdeglem4OLD 26009 elrspunidl 33157 |
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