Theorem fsuppeqg 8101

Description: Version of fsuppeq 8100 avoiding ax-rep 5212 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 8099	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍})))
2		ffun 6649	. . . . . 6 ⊢ (𝐹:𝐼⟶𝑆 → Fun 𝐹)
3		inpreima 6992	. . . . . 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 6026	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
6		fdm 6655	. . . . . . . 8 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = 𝐼)
7		fimacnv 6668	. . . . . . . 8 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ 𝑆) = 𝐼)
8	6, 7	eqtr4d 2769	. . . . . . 7 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = (◡𝐹 “ 𝑆))
9	5, 8	sseqtrid 3972	. . . . . 6 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡𝐹 “ 𝑆))
10		sseqin2 4168	. . . . . 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 2766	. . . 4 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍})))
13		invdif 4224	. . . . 5 ⊢ (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
14	13	imaeq2i 6002	. . . 4 ⊢ (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (𝑆 ∖ {𝑍}))
15	12, 14	eqtr3di 2781	. . 3 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) = (◡𝐹 “ (𝑆 ∖ {𝑍})))
16	1, 15	sylan9eq 2786	. 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 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3894   ∩ cin 3896   ⊆ wss 3897  {csn 4571  ^◡ccnv 5610  dom cdm 5611   “ cima 5614  Fun wfun 6470  ⟶wf 6472  (class class class)co 7341   supp csupp 8085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-supp 8086

This theorem is referenced by:  fcdmnn0suppg  12435