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Theorem fsuppeqg 8155
Description: Version of fsuppeq 8154 avoiding ax-rep 5275 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 8153 . . 3 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2 ffun 6710 . . . . . 6 (𝐹:𝐼𝑆 → Fun 𝐹)
3 inpreima 7055 . . . . . 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 6070 . . . . . . 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 2767 . . . . . . 7 (𝐹:𝐼𝑆 → dom 𝐹 = (𝐹𝑆))
95, 8sseqtrid 4026 . . . . . 6 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆))
10 sseqin2 4207 . . . . . 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 2764 . . . 4 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
13 invdif 4260 . . . . 5 (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
1413imaeq2i 6047 . . . 4 (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (𝑆 ∖ {𝑍}))
1512, 14eqtr3di 2779 . . 3 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
161, 15sylan9eq 2784 . 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 1533  wcel 2098  Vcvv 3466  cdif 3937  cin 3939  wss 3940  {csn 4620  ccnv 5665  dom cdm 5666  cima 5669  Fun wfun 6527  wf 6529  (class class class)co 7401   supp csupp 8140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-supp 8141
This theorem is referenced by:  fcdmnn0suppg  12526
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