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

Theorem frnsuppeq 7292
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 6475 . . . . . . 7 ((𝐹:𝐼𝑆𝐼𝑉) → 𝐹 ∈ V)
21expcom 451 . . . . . 6 (𝐼𝑉 → (𝐹:𝐼𝑆𝐹 ∈ V))
32adantr 481 . . . . 5 ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆𝐹 ∈ V))
43imp 445 . . . 4 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → 𝐹 ∈ V)
5 simplr 791 . . . 4 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → 𝑍𝑊)
6 suppimacnv 7291 . . . 4 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
74, 5, 6syl2anc 692 . . 3 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
8 invdif 3860 . . . . . 6 (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
98imaeq2i 5452 . . . . 5 (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (𝑆 ∖ {𝑍}))
10 ffun 6035 . . . . . . 7 (𝐹:𝐼𝑆 → Fun 𝐹)
11 inpreima 6328 . . . . . . 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 5473 . . . . . . . 8 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
14 fdm 6038 . . . . . . . . 9 (𝐹:𝐼𝑆 → dom 𝐹 = 𝐼)
15 fimacnv 6333 . . . . . . . . 9 (𝐹:𝐼𝑆 → (𝐹𝑆) = 𝐼)
1614, 15eqtr4d 2657 . . . . . . . 8 (𝐹:𝐼𝑆 → dom 𝐹 = (𝐹𝑆))
1713, 16syl5sseq 3645 . . . . . . 7 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆))
18 sseqin2 3809 . . . . . . 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 2654 . . . . 5 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
219, 20syl5reqr 2669 . . . 4 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
2221adantl 482 . . 3 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
237, 22eqtrd 2654 . 2 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍})))
2423ex 450 1 ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cdif 3564  cin 3566  wss 3567  {csn 4168  ccnv 5103  dom cdm 5104  cima 5107  Fun wfun 5870  wf 5872  (class class class)co 6635   supp csupp 7280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-supp 7281
This theorem is referenced by:  frnfsuppbi  8289  frnnn0supp  11334  ffs2  29477  eulerpartlemmf  30411  pwfi2f1o  37485
  Copyright terms: Public domain W3C validator