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Theorem suppimacnvss 7290
Description: The support of functions "defined" by inverse images is a subset of the support defined by df-supp 7281. (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 1793 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → ∃𝑦 𝑥𝑅𝑦)
2 pm5.1 901 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑍) → (𝑥𝑅𝑦𝑦𝑍))
32eximi 1760 . . . . 5 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
41, 3jca 554 . . . 4 (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
54a1i 11 . . 3 ((𝑅𝑉𝑍𝑊) → (∃𝑦(𝑥𝑅𝑦𝑦𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
65ss2abdv 3667 . 2 ((𝑅𝑉𝑍𝑊) → {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)} ⊆ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
7 cnvimadfsn 7289 . . 3 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
87a1i 11 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)})
9 suppvalbr 7284 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦𝑦𝑍))})
106, 8, 93sstr4d 3640 1 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wne 2791  Vcvv 3195  cdif 3564  wss 3567  {csn 4168   class class class wbr 4644  ccnv 5103  cima 5107  (class class class)co 6635   supp csupp 7280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-supp 7281
This theorem is referenced by:  suppimacnv  7291
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