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Theorem suppimacnvss 8160
Description: The support of functions "defined" by inverse images is a subset of the support defined by df-supp 8149. (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 1871 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → ∃𝑦 𝑥𝑅𝑦)
2 pm5.1 822 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑍) → (𝑥𝑅𝑦𝑦𝑍))
32eximi 1837 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
41, 3jca 512 . . . 4 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
54a1i 11 . . 3 ((𝑅𝑉𝑍𝑊) → (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
65ss2abdv 4060 . 2 ((𝑅𝑉𝑍𝑊) → {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)} ⊆ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
7 cnvimadfsn 8159 . . 3 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
87a1i 11 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)})
9 suppvalbr 8152 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
106, 8, 93sstr4d 4029 1 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wne 2940  Vcvv 3474  cdif 3945  wss 3948  {csn 4628   class class class wbr 5148  ccnv 5675  cima 5679  (class class class)co 7411   supp csupp 8148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-supp 8149
This theorem is referenced by:  suppimacnv  8161  fsuppinisegfi  31947
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