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Theorem strov2rcl 16534
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 16533 . 2 (𝑋𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
54simpld 495 1 (𝑋𝐵𝐼 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  dom cdm 5548  Rel wrel 5553  cfv 6348  (class class class)co 7145  Basecbs 16471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-slot 16475  df-base 16477
This theorem is referenced by:  mplrcl  20198  psropprmul  20334  dsmmbas2  20809  frlmrcl  20829
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