Theorem funisfsuppdc 7298

Description: The property of a function to be finitely supported (in relation to a given zero). This shows that df-fsuppfi 7293 and df-supp 6480 give the same results, in the case of decidable equality. (Contributed by AV, 23-May-2019.) (Revised by Jim Kingdon, 17-Oct-2026.)

Hypotheses

Ref	Expression
funisfsuppdc.r	⊢ (𝜑 → 𝐹 ∈ 𝑉)
funisfsuppdc.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
funisfsuppdc.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
funisfsuppdc.dmdc	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
funisfsuppdc.cdmdc	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)

Assertion

Ref	Expression
funisfsuppdc	⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑦,𝑍

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝑉(𝑥, 𝑦)   𝑍(𝑥)

Proof of Theorem funisfsuppdc

Dummy variables 𝑟 𝑝 𝑤 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funisfsuppdc.r	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		funisfsuppdc.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
3		isfsuppfi 7296	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝐹 finSupp 𝑍 ↔ (Fun 𝐹 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍}))))
4	1, 2, 3	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (Fun 𝐹 ∧ ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍}))))
5	4	simplbda 384	. . 3 ⊢ ((𝜑 ∧ 𝐹 finSupp 𝑍) → ∃𝑤(𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍}))
6		simpr1 1034	. . . 4 ⊢ (((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → 𝑤 ∈ Fin)
7		funisfsuppdc.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8	7	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → 𝐹:𝐴⟶𝐵)
9		simplr3 1072	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})
10		simpr 110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → 𝑞 ∈ (𝐴 ∖ 𝑤))
11	7	fdmd 5543	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝐴)
12	11	ad3antrrr 496	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → dom 𝐹 = 𝐴)
13	12	difeq1d 3346	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → (dom 𝐹 ∖ 𝑤) = (𝐴 ∖ 𝑤))
14	10, 13	eleqtrrd 2318	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → 𝑞 ∈ (dom 𝐹 ∖ 𝑤))
15	7	ffund 5540	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
16	15	ad3antrrr 496	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → Fun 𝐹)
17	10	eldifad 3231	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → 𝑞 ∈ 𝐴)
18	17, 12	eleqtrrd 2318	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → 𝑞 ∈ dom 𝐹)
19		funfvima 5953	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑞 ∈ dom 𝐹) → (𝑞 ∈ (dom 𝐹 ∖ 𝑤) → (𝐹‘𝑞) ∈ (𝐹 “ (dom 𝐹 ∖ 𝑤))))
20	16, 18, 19	syl2anc 415	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → (𝑞 ∈ (dom 𝐹 ∖ 𝑤) → (𝐹‘𝑞) ∈ (𝐹 “ (dom 𝐹 ∖ 𝑤))))
21	14, 20	mpd 13	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → (𝐹‘𝑞) ∈ (𝐹 “ (dom 𝐹 ∖ 𝑤)))
22	9, 21	sseldd 3249	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → (𝐹‘𝑞) ∈ {𝑍})
23		elsni 3727	. . . . . 6 ⊢ ((𝐹‘𝑞) ∈ {𝑍} → (𝐹‘𝑞) = 𝑍)
24	22, 23	syl 14	. . . . 5 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑞 ∈ (𝐴 ∖ 𝑤)) → (𝐹‘𝑞) = 𝑍)
25		funisfsuppdc.dmdc	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
26	25	ad3antrrr 496	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑟 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
27		simpr 110	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴)
28		simpr2 1035	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → 𝑤 ⊆ dom 𝐹)
29	11	sseq2d 3278	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ⊆ dom 𝐹 ↔ 𝑤 ⊆ 𝐴))
30	29	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → (𝑤 ⊆ dom 𝐹 ↔ 𝑤 ⊆ 𝐴))
31	28, 30	mpbid 147	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → 𝑤 ⊆ 𝐴)
32	31	adantr 276	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑟 ∈ 𝐴) → 𝑤 ⊆ 𝐴)
33	6	adantr 276	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑟 ∈ 𝐴) → 𝑤 ∈ Fin)
34	26, 27, 32, 33	elssdc 7215	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑟 ∈ 𝐴) → DECID 𝑟 ∈ 𝑤)
35	34	ralrimiva 2623	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → ∀𝑟 ∈ 𝐴 DECID 𝑟 ∈ 𝑤)
36	8, 24, 35	suppssdc 6504	. . . 4 ⊢ (((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → (𝐹 supp 𝑍) ⊆ 𝑤)
37		eqeq2 2248	. . . . . . . . 9 ⊢ (𝑦 = 𝑍 → ((𝐹‘𝑝) = 𝑦 ↔ (𝐹‘𝑝) = 𝑍))
38	37	dcbid 850	. . . . . . . 8 ⊢ (𝑦 = 𝑍 → (DECID (𝐹‘𝑝) = 𝑦 ↔ DECID (𝐹‘𝑝) = 𝑍))
39		eqeq1 2245	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑝) → (𝑥 = 𝑦 ↔ (𝐹‘𝑝) = 𝑦))
40	39	dcbid 850	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑝) → (DECID 𝑥 = 𝑦 ↔ DECID (𝐹‘𝑝) = 𝑦))
41	40	ralbidv 2550	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑝) → (∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐵 DECID (𝐹‘𝑝) = 𝑦))
42		funisfsuppdc.cdmdc	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)
43	42	ad3antrrr 496	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)
44	7	ad3antrrr 496	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → 𝐹:𝐴⟶𝐵)
45	31	sselda 3248	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → 𝑝 ∈ 𝐴)
46	44, 45	ffvelcdmd 5847	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → (𝐹‘𝑝) ∈ 𝐵)
47	41, 43, 46	rspcdva 2934	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → ∀𝑦 ∈ 𝐵 DECID (𝐹‘𝑝) = 𝑦)
48	2	ad3antrrr 496	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → 𝑍 ∈ 𝐵)
49	38, 47, 48	rspcdva 2934	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → DECID (𝐹‘𝑝) = 𝑍)
50	49	dcned 2426	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → DECID (𝐹‘𝑝) ≠ 𝑍)
51	7	ffnd 5537	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
52		elsuppfng 6486	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝑝 ∈ (𝐹 supp 𝑍) ↔ (𝑝 ∈ 𝐴 ∧ (𝐹‘𝑝) ≠ 𝑍)))
53	51, 1, 2, 52	syl3anc 1278	. . . . . . . . 9 ⊢ (𝜑 → (𝑝 ∈ (𝐹 supp 𝑍) ↔ (𝑝 ∈ 𝐴 ∧ (𝐹‘𝑝) ≠ 𝑍)))
54	53	ad3antrrr 496	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → (𝑝 ∈ (𝐹 supp 𝑍) ↔ (𝑝 ∈ 𝐴 ∧ (𝐹‘𝑝) ≠ 𝑍)))
55	45, 54	mpbirand 445	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → (𝑝 ∈ (𝐹 supp 𝑍) ↔ (𝐹‘𝑝) ≠ 𝑍))
56	55	dcbid 850	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → (DECID 𝑝 ∈ (𝐹 supp 𝑍) ↔ DECID (𝐹‘𝑝) ≠ 𝑍))
57	50, 56	mpbird 167	. . . . 5 ⊢ ((((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) ∧ 𝑝 ∈ 𝑤) → DECID 𝑝 ∈ (𝐹 supp 𝑍))
58	57	ralrimiva 2623	. . . 4 ⊢ (((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → ∀𝑝 ∈ 𝑤 DECID 𝑝 ∈ (𝐹 supp 𝑍))
59		ssfidc 7251	. . . 4 ⊢ ((𝑤 ∈ Fin ∧ (𝐹 supp 𝑍) ⊆ 𝑤 ∧ ∀𝑝 ∈ 𝑤 DECID 𝑝 ∈ (𝐹 supp 𝑍)) → (𝐹 supp 𝑍) ∈ Fin)
60	6, 36, 58, 59	syl3anc 1278	. . 3 ⊢ (((𝜑 ∧ 𝐹 finSupp 𝑍) ∧ (𝑤 ∈ Fin ∧ 𝑤 ⊆ dom 𝐹 ∧ (𝐹 “ (dom 𝐹 ∖ 𝑤)) ⊆ {𝑍})) → (𝐹 supp 𝑍) ∈ Fin)
61	5, 60	exlimddv 1954	. 2 ⊢ ((𝜑 ∧ 𝐹 finSupp 𝑍) → (𝐹 supp 𝑍) ∈ Fin)
62	1	adantr 276	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 𝑍) ∈ Fin) → 𝐹 ∈ 𝑉)
63	2	adantr 276	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 𝑍) ∈ Fin) → 𝑍 ∈ 𝐵)
64	15	adantr 276	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 𝑍) ∈ Fin) → Fun 𝐹)
65		simpr 110	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 𝑍) ∈ Fin) → (𝐹 supp 𝑍) ∈ Fin)
66		suppssdmg 6493	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝐵) → (𝐹 supp 𝑍) ⊆ dom 𝐹)
67	62, 63, 66	syl2anc 415	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 𝑍) ∈ Fin) → (𝐹 supp 𝑍) ⊆ dom 𝐹)
68	11	difeq1d 3346	. . . . . 6 ⊢ (𝜑 → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (𝐴 ∖ (𝐹 supp 𝑍)))
69	68	imaeq2d 5129	. . . . 5 ⊢ (𝜑 → (𝐹 “ (dom 𝐹 ∖ (𝐹 supp 𝑍))) = (𝐹 “ (𝐴 ∖ (𝐹 supp 𝑍))))
70	7, 1, 2, 42	suppinisegdc 6513	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))
71	70	imaeq2d 5129	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ (𝐴 ∖ (𝐹 supp 𝑍))) = (𝐹 “ (◡𝐹 “ {𝑍})))
72		funimacnv 5460	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
73	15, 72	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
74	71, 73	eqtrd 2271	. . . . . 6 ⊢ (𝜑 → (𝐹 “ (𝐴 ∖ (𝐹 supp 𝑍))) = ({𝑍} ∩ ran 𝐹))
75		inss1 3451	. . . . . 6 ⊢ ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
76	74, 75	eqsstrdi 3300	. . . . 5 ⊢ (𝜑 → (𝐹 “ (𝐴 ∖ (𝐹 supp 𝑍))) ⊆ {𝑍})
77	69, 76	eqsstrd 3284	. . . 4 ⊢ (𝜑 → (𝐹 “ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ {𝑍})
78	77	adantr 276	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 𝑍) ∈ Fin) → (𝐹 “ (dom 𝐹 ∖ (𝐹 supp 𝑍))) ⊆ {𝑍})
79	62, 63, 64, 65, 67, 78	isfsuppfid 7297	. 2 ⊢ ((𝜑 ∧ (𝐹 supp 𝑍) ∈ Fin) → 𝐹 finSupp 𝑍)
80	61, 79	impbida 604	1 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528   ∖ cdif 3217   ∩ cin 3219   ⊆ wss 3220  {csn 3709   class class class wbr 4131  ^◡ccnv 4775  dom cdm 4776  ran crn 4777   “ cima 4779  Fun wfun 5374   Fn wfn 5375  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088   supp csupp 6479  Fincfn 7028   finSupp cfsupp 7292

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-supp 6480  df-1o 6692  df-er 6812  df-en 7029  df-fin 7031  df-fsuppfi 7293

This theorem is used by:  fcdmnn0fsuppfi  9742