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Theorem fsuppeqg 8201
Description: Version of fsuppeq 8200 avoiding ax-rep 5279 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 30-Jul-2024.)
Assertion
Ref Expression
fsuppeqg ((𝐹𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))

Proof of Theorem fsuppeqg
StepHypRef Expression
1 suppimacnv 8199 . . 3 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2 ffun 6739 . . . . . 6 (𝐹:𝐼𝑆 → Fun 𝐹)
3 inpreima 7084 . . . . . 6 (Fun 𝐹 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))))
42, 3syl 17 . . . . 5 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))))
5 cnvimass 6100 . . . . . . 7 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
6 fdm 6745 . . . . . . . 8 (𝐹:𝐼𝑆 → dom 𝐹 = 𝐼)
7 fimacnv 6758 . . . . . . . 8 (𝐹:𝐼𝑆 → (𝐹𝑆) = 𝐼)
86, 7eqtr4d 2780 . . . . . . 7 (𝐹:𝐼𝑆 → dom 𝐹 = (𝐹𝑆))
95, 8sseqtrid 4026 . . . . . 6 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆))
10 sseqin2 4223 . . . . . 6 ((𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆) ↔ ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
119, 10sylib 218 . . . . 5 (𝐹:𝐼𝑆 → ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
124, 11eqtrd 2777 . . . 4 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
13 invdif 4279 . . . . 5 (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
1413imaeq2i 6076 . . . 4 (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (𝑆 ∖ {𝑍}))
1512, 14eqtr3di 2792 . . 3 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
161, 15sylan9eq 2797 . 2 (((𝐹𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍})))
1716ex 412 1 ((𝐹𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))
Colors of variables: wff setvar class
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-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-rab 3437  df-v 3482  df-sbc 3789  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-br 5144  df-opab 5206  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-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  fcdmnn0suppg  12585
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