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Theorem fsuppeq 8144
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
fsuppeq ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))

Proof of Theorem fsuppeq
StepHypRef Expression
1 fex 7213 . . . . . . 7 ((𝐹:𝐼𝑆𝐼𝑉) → 𝐹 ∈ V)
21expcom 414 . . . . . 6 (𝐼𝑉 → (𝐹:𝐼𝑆𝐹 ∈ V))
32adantr 481 . . . . 5 ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆𝐹 ∈ V))
43imp 407 . . . 4 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → 𝐹 ∈ V)
5 simplr 767 . . . 4 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → 𝑍𝑊)
6 suppimacnv 8143 . . . 4 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
74, 5, 6syl2anc 584 . . 3 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
8 ffun 6708 . . . . . . 7 (𝐹:𝐼𝑆 → Fun 𝐹)
9 inpreima 7051 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))))
108, 9syl 17 . . . . . 6 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))))
11 cnvimass 6070 . . . . . . . 8 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
12 fdm 6714 . . . . . . . . 9 (𝐹:𝐼𝑆 → dom 𝐹 = 𝐼)
13 fimacnv 6727 . . . . . . . . 9 (𝐹:𝐼𝑆 → (𝐹𝑆) = 𝐼)
1412, 13eqtr4d 2775 . . . . . . . 8 (𝐹:𝐼𝑆 → dom 𝐹 = (𝐹𝑆))
1511, 14sseqtrid 4031 . . . . . . 7 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆))
16 sseqin2 4212 . . . . . . 7 ((𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆) ↔ ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
1715, 16sylib 217 . . . . . 6 (𝐹:𝐼𝑆 → ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
1810, 17eqtrd 2772 . . . . 5 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
19 invdif 4265 . . . . . 6 (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
2019imaeq2i 6048 . . . . 5 (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (𝑆 ∖ {𝑍}))
2118, 20eqtr3di 2787 . . . 4 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
2221adantl 482 . . 3 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
237, 22eqtrd 2772 . 2 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍})))
2423ex 413 1 ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cdif 3942  cin 3944  wss 3945  {csn 4623  ccnv 5669  dom cdm 5670  cima 5673  Fun wfun 6527  wf 6529  (class class class)co 7394   supp csupp 8130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-supp 8131
This theorem is referenced by:  ffsuppbi  9377  fcdmnn0supp  12512  mhpmulcl  21623  ffs2  31888  eulerpartlemmf  33269  pwfi2f1o  41673
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