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

Theorem frnsuppeq 7194
 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 6394 . . . . . . 7 ((𝐹:𝐼𝑆𝐼𝑉) → 𝐹 ∈ V)
21expcom 450 . . . . . 6 (𝐼𝑉 → (𝐹:𝐼𝑆𝐹 ∈ V))
32adantr 480 . . . . 5 ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆𝐹 ∈ V))
43imp 444 . . . 4 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → 𝐹 ∈ V)
5 simplr 788 . . . 4 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → 𝑍𝑊)
6 suppimacnv 7193 . . . 4 ((𝐹 ∈ V ∧ 𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
74, 5, 6syl2anc 691 . . 3 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
8 invdif 3827 . . . . . 6 (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍})
98imaeq2i 5383 . . . . 5 (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (𝑆 ∖ {𝑍}))
10 ffun 5961 . . . . . . 7 (𝐹:𝐼𝑆 → Fun 𝐹)
11 inpreima 6250 . . . . . . 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 5404 . . . . . . . 8 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
14 fdm 5964 . . . . . . . . 9 (𝐹:𝐼𝑆 → dom 𝐹 = 𝐼)
15 fimacnv 6255 . . . . . . . . 9 (𝐹:𝐼𝑆 → (𝐹𝑆) = 𝐼)
1614, 15eqtr4d 2647 . . . . . . . 8 (𝐹:𝐼𝑆 → dom 𝐹 = (𝐹𝑆))
1713, 16syl5sseq 3616 . . . . . . 7 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆))
18 sseqin2 3779 . . . . . . 7 ((𝐹 “ (V ∖ {𝑍})) ⊆ (𝐹𝑆) ↔ ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
1917, 18sylib 207 . . . . . 6 (𝐹:𝐼𝑆 → ((𝐹𝑆) ∩ (𝐹 “ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
2012, 19eqtrd 2644 . . . . 5 (𝐹:𝐼𝑆 → (𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (𝐹 “ (V ∖ {𝑍})))
219, 20syl5reqr 2659 . . . 4 (𝐹:𝐼𝑆 → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
2221adantl 481 . . 3 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 “ (V ∖ {𝑍})) = (𝐹 “ (𝑆 ∖ {𝑍})))
237, 22eqtrd 2644 . 2 (((𝐼𝑉𝑍𝑊) ∧ 𝐹:𝐼𝑆) → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍})))
2423ex 449 1 ((𝐼𝑉𝑍𝑊) → (𝐹:𝐼𝑆 → (𝐹 supp 𝑍) = (𝐹 “ (𝑆 ∖ {𝑍}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ◡ccnv 5037  dom cdm 5038   “ cima 5041  Fun wfun 5798  ⟶wf 5800  (class class class)co 6549   supp csupp 7182 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-supp 7183 This theorem is referenced by:  frnfsuppbi  8187  frnnn0supp  11226  ffs2  28891  eulerpartlemmf  29764  pwfi2f1o  36684
 Copyright terms: Public domain W3C validator