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Theorem elbasov 16401
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 4185 . 2 (𝐴𝐵 → ¬ 𝐵 = ∅)
2 elbasov.s . . . . 5 𝑆 = (𝑋𝑂𝑌)
3 elbasov.o . . . . . 6 Rel dom 𝑂
43ovprc 7013 . . . . 5 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅)
52, 4syl5eq 2826 . . . 4 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑆 = ∅)
65fveq2d 6503 . . 3 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (Base‘𝑆) = (Base‘∅))
7 elbasov.b . . 3 𝐵 = (Base‘𝑆)
8 base0 16392 . . 3 ∅ = (Base‘∅)
96, 7, 83eqtr4g 2839 . 2 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝐵 = ∅)
101, 9nsyl2 145 1 (𝐴𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1507  wcel 2050  Vcvv 3415  c0 4178  dom cdm 5407  Rel wrel 5412  cfv 6188  (class class class)co 6976  Basecbs 16339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-slot 16343  df-base 16345
This theorem is referenced by:  strov2rcl  16402  psrelbas  19873  psraddcl  19877  psrmulcllem  19881  psrvscafval  19884  psrvscacl  19887  resspsradd  19910  resspsrmul  19911  cphsubrglem  23484  mdegcl  24366
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