Theorem fsuppeq 8230

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 7263	. . . . . . 7 ⊢ ((𝐹:𝐼⟶𝑆 ∧ 𝐼 ∈ 𝑉) → 𝐹 ∈ V)
2	1	expcom 419	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐹:𝐼⟶𝑆 → 𝐹 ∈ V))
3	2	adantr 486	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → 𝐹 ∈ V))
4	3	imp 412	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → 𝐹 ∈ V)
5		simplr 781	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → 𝑍 ∈ 𝑊)
6		suppimacnv 8229	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
7	4, 5, 6	syl2anc 596	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
8		ffun 6742	. . . . . . 7 ⊢ (𝐹:𝐼⟶𝑆 → Fun 𝐹)
9		inpreima 7095	. . . . . . 7 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))))
10	8, 9	syl 18	. . . . . 6 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))))
11		cnvimass 6217	. . . . . . . 8 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
12		fdm 6749	. . . . . . . . 9 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = 𝐼)
13		fimacnv 6762	. . . . . . . . 9 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ 𝑆) = 𝐼)
14	12, 13	eqtr4d 2801	. . . . . . . 8 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = (◡𝐹 “ 𝑆))
15	11, 14	sseqtrid 3955	. . . . . . 7 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡𝐹 “ 𝑆))
16		sseqin2 4123	. . . . . . 7 ⊢ ((◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡𝐹 “ 𝑆) ↔ ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
17	15, 16	sylib 221	. . . . . 6 ⊢ (𝐹:𝐼⟶𝑆 → ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
18	10, 17	eqtrd 2798	. . . . 5 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
19		invdif 4227	. . . . . 6 ⊢ (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
20	19	imaeq2i 6068	. . . . 5 ⊢ (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (𝑆 ∖ {𝑍}))
21	18, 20	eqtr3di 2813	. . . 4 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
22	21	adantl 487	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (◡𝐹 “ (V ∖ {𝑍})) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
23	7, 22	eqtrd 2798	. 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
24	23	ex 418	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  {csn 4588  ^◡ccnv 5657  dom cdm 5658   “ cima 5661  Fun wfun 6560  ⟶wf 6562  (class class class)co 7451   supp csupp 8214

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-supp 8215

This theorem is used by:  ffsuppbi  9427  fcdmnn0supp  12703  mhpmulcl  22525  ffs2  33415  indsupp  33530  indfsid  33532  esplysply  34314  eulerpartlemmf  35118  pwfi2f1o  44364