MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsuppeq Structured version   Visualization version   GIF version

Theorem fsuppeq 8198
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 7245 . . . . . . 7 ((𝐹:𝐼𝑆𝐼𝑉) → 𝐹 ∈ V)
21expcom 413 . . . . . 6 (𝐼𝑉 → (𝐹:𝐼𝑆𝐹 ∈ V))
32adantr 480 . . . . 5 ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆𝐹 ∈ V))
43imp 406 . . . 4 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → 𝐹 ∈ V)
5 simplr 769 . . . 4 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → 𝑍𝑊)
6 suppimacnv 8197 . . . 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 6739 . . . . . . 7 (𝐹:𝐼𝑆 → Fun 𝐹)
9 inpreima 7083 . . . . . . 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 6101 . . . . . . . 8 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
12 fdm 6745 . . . . . . . . 9 (𝐹:𝐼𝑆 → dom 𝐹 = 𝐼)
13 fimacnv 6758 . . . . . . . . 9 (𝐹:𝐼𝑆 → (𝐹𝑆) = 𝐼)
1412, 13eqtr4d 2777 . . . . . . . 8 (𝐹:𝐼𝑆 → dom 𝐹 = (𝐹𝑆))
1511, 14sseqtrid 4047 . . . . . . 7 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆))
16 sseqin2 4230 . . . . . . 7 ((𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆) ↔ ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
1715, 16sylib 218 . . . . . 6 (𝐹:𝐼𝑆 → ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
1810, 17eqtrd 2774 . . . . 5 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
19 invdif 4284 . . . . . 6 (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
2019imaeq2i 6077 . . . . 5 (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (𝑆 ∖ {𝑍}))
2118, 20eqtr3di 2789 . . . 4 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
2221adantl 481 . . 3 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
237, 22eqtrd 2774 . 2 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍})))
2423ex 412 1 ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  cdif 3959  cin 3961  wss 3962  {csn 4630  ccnv 5687  dom cdm 5688  cima 5691  Fun wfun 6556  wf 6558  (class class class)co 7430   supp csupp 8183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-supp 8184
This theorem is referenced by:  ffsuppbi  9435  fcdmnn0supp  12580  mhpmulcl  22170  ffs2  32745  eulerpartlemmf  34356  pwfi2f1o  43084
  Copyright terms: Public domain W3C validator