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Theorem suppimacnvss 7827
 Description: The support of functions "defined" by inverse images is a subset of the support defined by df-supp 7818. (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 1869 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → ∃𝑦 𝑥𝑅𝑦)
2 pm5.1 822 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑍) → (𝑥𝑅𝑦𝑦𝑍))
32eximi 1836 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
41, 3jca 515 . . . 4 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
54a1i 11 . . 3 ((𝑅𝑉𝑍𝑊) → (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
65ss2abdv 3994 . 2 ((𝑅𝑉𝑍𝑊) → {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)} ⊆ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
7 cnvimadfsn 7826 . . 3 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
87a1i 11 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)})
9 suppvalbr 7821 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
106, 8, 93sstr4d 3965 1 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112  {cab 2779   ≠ wne 2990  Vcvv 3444   ∖ cdif 3881   ⊆ wss 3884  {csn 4528   class class class wbr 5033  ◡ccnv 5522   “ cima 5526  (class class class)co 7139   supp csupp 7817 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-supp 7818 This theorem is referenced by:  suppimacnv  7828  fsuppinisegfi  30450
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