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

Theorem suppimacnvss 8161
Description: The support of functions "defined" by inverse images is a subset of the support defined by df-supp 8150. (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 1870 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → ∃𝑦 𝑥𝑅𝑦)
2 pm5.1 821 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑍) → (𝑥𝑅𝑦𝑦𝑍))
32eximi 1836 . . . . 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 4061 . 2 ((𝑅𝑉𝑍𝑊) → {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)} ⊆ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
7 cnvimadfsn 8160 . . 3 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
87a1i 11 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)})
9 suppvalbr 8153 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
106, 8, 93sstr4d 4030 1 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wex 1780  wcel 2105  {cab 2708  wne 2939  Vcvv 3473  cdif 3946  wss 3949  {csn 4629   class class class wbr 5149  ccnv 5676  cima 5680  (class class class)co 7412   supp csupp 8149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-supp 8150
This theorem is referenced by:  suppimacnv  8162  fsuppinisegfi  32173
  Copyright terms: Public domain W3C validator