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Theorem frnsuppeq 7458
Description: Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
Assertion
Ref Expression
frnsuppeq ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))

Proof of Theorem frnsuppeq
StepHypRef Expression
1 fex 6633 . . . . . . 7 ((𝐹:𝐼𝑆𝐼𝑉) → 𝐹 ∈ V)
21expcom 398 . . . . . 6 (𝐼𝑉 → (𝐹:𝐼𝑆𝐹 ∈ V))
32adantr 466 . . . . 5 ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆𝐹 ∈ V))
43imp 393 . . . 4 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → 𝐹 ∈ V)
5 simplr 752 . . . 4 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → 𝑍𝑊)
6 suppimacnv 7457 . . . 4 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
74, 5, 6syl2anc 573 . . 3 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
8 invdif 4017 . . . . . 6 (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
98imaeq2i 5605 . . . . 5 (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (𝑆 ∖ {𝑍}))
10 ffun 6188 . . . . . . 7 (𝐹:𝐼𝑆 → Fun 𝐹)
11 inpreima 6485 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))))
1210, 11syl 17 . . . . . 6 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))))
13 cnvimass 5626 . . . . . . . 8 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
14 fdm 6191 . . . . . . . . 9 (𝐹:𝐼𝑆 → dom 𝐹 = 𝐼)
15 fimacnv 6490 . . . . . . . . 9 (𝐹:𝐼𝑆 → (𝐹𝑆) = 𝐼)
1614, 15eqtr4d 2808 . . . . . . . 8 (𝐹:𝐼𝑆 → dom 𝐹 = (𝐹𝑆))
1713, 16syl5sseq 3802 . . . . . . 7 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆))
18 sseqin2 3968 . . . . . . 7 ((𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆) ↔ ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
1917, 18sylib 208 . . . . . 6 (𝐹:𝐼𝑆 → ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
2012, 19eqtrd 2805 . . . . 5 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
219, 20syl5reqr 2820 . . . 4 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
2221adantl 467 . . 3 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
237, 22eqtrd 2805 . 2 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍})))
2423ex 397 1 ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  cin 3722  wss 3723  {csn 4316  ccnv 5248  dom cdm 5249  cima 5252  Fun wfun 6025  wf 6027  (class class class)co 6793   supp csupp 7446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-supp 7447
This theorem is referenced by:  frnfsuppbi  8460  frnnn0supp  11551  ffs2  29843  eulerpartlemmf  30777  pwfi2f1o  38192
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