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

Theorem suppimacnvss 8199
Description: The support of functions "defined" by inverse images is a subset of the support defined by df-supp 8187. (Contributed by AV, 7-Apr-2019.)
Assertion
Ref Expression
suppimacnvss ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))

Proof of Theorem suppimacnvss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exsimpl 1867 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → ∃𝑦 𝑥𝑅𝑦)
2 pm5.1 823 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑍) → (𝑥𝑅𝑦𝑦𝑍))
32eximi 1834 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
41, 3jca 511 . . . 4 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
54a1i 11 . . 3 ((𝑅𝑉𝑍𝑊) → (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
65ss2abdv 4065 . 2 ((𝑅𝑉𝑍𝑊) → {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)} ⊆ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
7 cnvimadfsn 8198 . . 3 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
87a1i 11 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)})
9 suppvalbr 8190 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
106, 8, 93sstr4d 4038 1 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wne 2939  Vcvv 3479  cdif 3947  wss 3950  {csn 4625   class class class wbr 5142  ccnv 5683  cima 5687  (class class class)co 7432   supp csupp 8186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-supp 8187
This theorem is referenced by:  suppimacnv  8200  fsuppinisegfi  32697
  Copyright terms: Public domain W3C validator