Theorem fsuppeqg 8200

Description: Version of fsuppeq 8199 avoiding ax-rep 5285 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 30-Jul-2024.)

Assertion

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

Proof of Theorem fsuppeqg

Step	Hyp	Ref	Expression
1		suppimacnv 8198	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
2		ffun 6740	. . . . . 6 ⊢ (𝐹:𝐼⟶𝑆 → Fun 𝐹)
3		inpreima 7084	. . . . . 6 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))))
4	2, 3	syl 17	. . . . 5 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))))
5		cnvimass 6102	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
6		fdm 6746	. . . . . . . 8 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = 𝐼)
7		fimacnv 6759	. . . . . . . 8 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ 𝑆) = 𝐼)
8	6, 7	eqtr4d 2778	. . . . . . 7 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = (◡𝐹 “ 𝑆))
9	5, 8	sseqtrid 4048	. . . . . 6 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡𝐹 “ 𝑆))
10		sseqin2 4231	. . . . . 6 ⊢ ((◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡𝐹 “ 𝑆) ↔ ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
11	9, 10	sylib 218	. . . . 5 ⊢ (𝐹:𝐼⟶𝑆 → ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
12	4, 11	eqtrd 2775	. . . 4 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
13		invdif 4285	. . . . 5 ⊢ (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
14	13	imaeq2i 6078	. . . 4 ⊢ (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (𝑆 ∖ {𝑍}))
15	12, 14	eqtr3di 2790	. . 3 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
16	1, 15	sylan9eq 2795	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
17	16	ex 412	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  {csn 4631  ^◡ccnv 5688  dom cdm 5689   “ cima 5692  Fun wfun 6557  ⟶wf 6559  (class class class)co 7431   supp csupp 8184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8185

This theorem is referenced by:  fcdmnn0suppg  12583