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Theorem 0fsupp 8297

Description: The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.)

**Assertion**
Ref	Expression
0fsupp	⊢ (𝑍 ∈ 𝑉 → ∅ finSupp 𝑍)

**Proof of Theorem 0fsupp**
Step	Hyp	Ref	Expression
1		supp0 7300	. . 3 ⊢ (𝑍 ∈ 𝑉 → (∅ supp 𝑍) = ∅)
2		0fin 8188	. . 3 ⊢ ∅ ∈ Fin
3	1, 2	syl6eqel 2709	. 2 ⊢ (𝑍 ∈ 𝑉 → (∅ supp 𝑍) ∈ Fin)
4		fun0 5954	. . 3 ⊢ Fun ∅
5		0ex 4790	. . 3 ⊢ ∅ ∈ V
6		funisfsupp 8280	. . 3 ⊢ ((Fun ∅ ∧ ∅ ∈ V ∧ 𝑍 ∈ 𝑉) → (∅ finSupp 𝑍 ↔ (∅ supp 𝑍) ∈ Fin))
7	4, 5, 6	mp3an12 1414	. 2 ⊢ (𝑍 ∈ 𝑉 → (∅ finSupp 𝑍 ↔ (∅ supp 𝑍) ∈ Fin))
8	3, 7	mpbird 247	1 ⊢ (𝑍 ∈ 𝑉 → ∅ finSupp 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 class class class wbr 4653 Fun wfun 5882 (class class class)co 6650 supp csupp 7295 Fincfn 7955 finSupp cfsupp 8275

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-supp 7296 df-en 7956 df-fin 7959 df-fsupp 8276

This theorem is referenced by: lco0 42216

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