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Theorem strov2rcl 15750
 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 15749 . 2 (𝑋𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
54simpld 474 1 (𝑋𝐵𝐼 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  dom cdm 5038  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  Basecbs 15695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-slot 15699  df-base 15700 This theorem is referenced by:  mplrcl  19311  psropprmul  19429  frlmrcl  19920
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