Theorem fsuppeq 8200

Description: Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)

Assertion

Ref	Expression
fsuppeq	⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))))

Proof of Theorem fsuppeq

Step	Hyp	Ref	Expression
1		fex 7246	. . . . . . 7 ⊢ ((𝐹:𝐼⟶𝑆 ∧ 𝐼 ∈ 𝑉) → 𝐹 ∈ V)
2	1	expcom 413	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐹:𝐼⟶𝑆 → 𝐹 ∈ V))
3	2	adantr 480	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → 𝐹 ∈ V))
4	3	imp 406	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → 𝐹 ∈ V)
5		simplr 769	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → 𝑍 ∈ 𝑊)
6		suppimacnv 8199	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
7	4, 5, 6	syl2anc 584	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
8		ffun 6739	. . . . . . 7 ⊢ (𝐹:𝐼⟶𝑆 → Fun 𝐹)
9		inpreima 7084	. . . . . . 7 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))))
10	8, 9	syl 17	. . . . . 6 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))))
11		cnvimass 6100	. . . . . . . 8 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
12		fdm 6745	. . . . . . . . 9 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = 𝐼)
13		fimacnv 6758	. . . . . . . . 9 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ 𝑆) = 𝐼)
14	12, 13	eqtr4d 2780	. . . . . . . 8 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = (◡𝐹 “ 𝑆))
15	11, 14	sseqtrid 4026	. . . . . . 7 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡𝐹 “ 𝑆))
16		sseqin2 4223	. . . . . . 7 ⊢ ((◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡𝐹 “ 𝑆) ↔ ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
17	15, 16	sylib 218	. . . . . 6 ⊢ (𝐹:𝐼⟶𝑆 → ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
18	10, 17	eqtrd 2777	. . . . 5 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
19		invdif 4279	. . . . . 6 ⊢ (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
20	19	imaeq2i 6076	. . . . 5 ⊢ (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (𝑆 ∖ {𝑍}))
21	18, 20	eqtr3di 2792	. . . 4 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
22	21	adantl 481	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (◡𝐹 “ (V ∖ {𝑍})) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
23	7, 22	eqtrd 2777	. 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
24	23	ex 412	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  {csn 4626  ^◡ccnv 5684  dom cdm 5685   “ cima 5688  Fun wfun 6555  ⟶wf 6557  (class class class)co 7431   supp csupp 8185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186

This theorem is referenced by:  ffsuppbi  9438  fcdmnn0supp  12583  mhpmulcl  22153  ffs2  32739  indsupp  32852  eulerpartlemmf  34377  pwfi2f1o  43108