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Theorem fsuppeqg 8148
Description: Version of fsuppeq 8147 avoiding ax-rep 5281 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 8146 . . 3 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2 ffun 6710 . . . . . 6 (𝐹:𝐼𝑆 → Fun 𝐹)
3 inpreima 7053 . . . . . 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 6072 . . . . . . 7 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
6 fdm 6716 . . . . . . . 8 (𝐹:𝐼𝑆 → dom 𝐹 = 𝐼)
7 fimacnv 6729 . . . . . . . 8 (𝐹:𝐼𝑆 → (𝐹𝑆) = 𝐼)
86, 7eqtr4d 2776 . . . . . . 7 (𝐹:𝐼𝑆 → dom 𝐹 = (𝐹𝑆))
95, 8sseqtrid 4032 . . . . . 6 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆))
10 sseqin2 4213 . . . . . 6 ((𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆) ↔ ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
119, 10sylib 217 . . . . 5 (𝐹:𝐼𝑆 → ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
124, 11eqtrd 2773 . . . 4 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
13 invdif 4266 . . . . 5 (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
1413imaeq2i 6050 . . . 4 (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (𝑆 ∖ {𝑍}))
1512, 14eqtr3di 2788 . . 3 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
161, 15sylan9eq 2793 . 2 (((𝐹𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍})))
1716ex 414 1 ((𝐹𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cdif 3943  cin 3945  wss 3946  {csn 4624  ccnv 5671  dom cdm 5672  cima 5675  Fun wfun 6529  wf 6531  (class class class)co 7396   supp csupp 8133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-supp 8134
This theorem is referenced by:  fcdmnn0suppg  12517
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