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Theorem elbasov 15749
 Description: Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.)
Hypotheses
Ref Expression
elbasov.o Rel dom 𝑂
elbasov.s 𝑆 = (𝑋𝑂𝑌)
elbasov.b 𝐵 = (Base‘𝑆)
Assertion
Ref Expression
elbasov (𝐴𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V))

Proof of Theorem elbasov
StepHypRef Expression
1 n0i 3879 . 2 (𝐴𝐵 → ¬ 𝐵 = ∅)
2 elbasov.s . . . . 5 𝑆 = (𝑋𝑂𝑌)
3 elbasov.o . . . . . 6 Rel dom 𝑂
43ovprc 6581 . . . . 5 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅)
52, 4syl5eq 2656 . . . 4 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑆 = ∅)
65fveq2d 6107 . . 3 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (Base‘𝑆) = (Base‘∅))
7 elbasov.b . . 3 𝐵 = (Base‘𝑆)
8 base0 15740 . . 3 ∅ = (Base‘∅)
96, 7, 83eqtr4g 2669 . 2 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝐵 = ∅)
101, 9nsyl2 141 1 (𝐴𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  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:  strov2rcl  15750  psrelbas  19200  psraddcl  19204  psrmulcllem  19208  psrvscafval  19211  psrvscacl  19214  resspsradd  19237  resspsrmul  19238  cphsubrglem  22785  mdegcl  23633
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