Theorem fsuppeq 8170

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 7225	. . . . . . 7 ⊢ ((𝐹:𝐼⟶𝑆 ∧ 𝐼 ∈ 𝑉) → 𝐹 ∈ V)
2	1	expcom 418	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐹:𝐼⟶𝑆 → 𝐹 ∈ V))
3	2	adantr 485	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → 𝐹 ∈ V))
4	3	imp 411	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → 𝐹 ∈ V)
5		simplr 780	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → 𝑍 ∈ 𝑊)
6		suppimacnv 8169	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
7	4, 5, 6	syl2anc 595	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
8		ffun 6709	. . . . . . 7 ⊢ (𝐹:𝐼⟶𝑆 → Fun 𝐹)
9		inpreima 7060	. . . . . . 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 6085	. . . . . . . 8 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
12		fdm 6716	. . . . . . . . 9 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = 𝐼)
13		fimacnv 6729	. . . . . . . . 9 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ 𝑆) = 𝐼)
14	12, 13	eqtr4d 2807	. . . . . . . 8 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = (◡𝐹 “ 𝑆))
15	11, 14	sseqtrid 3987	. . . . . . 7 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡𝐹 “ 𝑆))
16		sseqin2 4184	. . . . . . 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 2804	. . . . 5 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
19		invdif 4240	. . . . . 6 ⊢ (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
20	19	imaeq2i 6061	. . . . 5 ⊢ (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (𝑆 ∖ {𝑍}))
21	18, 20	eqtr3di 2819	. . . 4 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
22	21	adantl 486	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (◡𝐹 “ (V ∖ {𝑍})) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
23	7, 22	eqtrd 2804	. 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
24	23	ex 417	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  {csn 4594  ^◡ccnv 5661  dom cdm 5662   “ cima 5665  Fun wfun 6531  ⟶wf 6533  (class class class)co 7411   supp csupp 8155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-supp 8156

This theorem is referenced by:  ffsuppbi  9357  fcdmnn0supp  12560  mhpmulcl  22280  ffs2  33012  indsupp  33127  indfsid  33129  esplysply  33905  eulerpartlemmf  34709  pwfi2f1o  43714