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Theorem strov2rcl 16537
Description: Partial reverse closure for any structure defined as a two-argument function. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 2-Dec-2019.)
Hypotheses
Ref Expression
strov2rcl.s 𝑆 = (𝐼𝐹𝑅)
strov2rcl.b 𝐵 = (Base‘𝑆)
strov2rcl.f Rel dom 𝐹
Assertion
Ref Expression
strov2rcl (𝑋𝐵𝐼 ∈ V)

Proof of Theorem strov2rcl
StepHypRef Expression
1 strov2rcl.f . . 3 Rel dom 𝐹
2 strov2rcl.s . . 3 𝑆 = (𝐼𝐹𝑅)
3 strov2rcl.b . . 3 𝐵 = (Base‘𝑆)
41, 2, 3elbasov 16536 . 2 (𝑋𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
54simpld 498 1 (𝑋𝐵𝐼 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  Vcvv 3469  dom cdm 5532  Rel wrel 5537  cfv 6334  (class class class)co 7140  Basecbs 16474
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-slot 16478  df-base 16480
This theorem is referenced by:  dsmmbas2  20424  frlmrcl  20444  mplrcl  20727  psropprmul  20865
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