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Theorem elbasov 15842
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 3896 . 2 (𝐴𝐵 → ¬ 𝐵 = ∅)
2 elbasov.s . . . . 5 𝑆 = (𝑋𝑂𝑌)
3 elbasov.o . . . . . 6 Rel dom 𝑂
43ovprc 6636 . . . . 5 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅)
52, 4syl5eq 2667 . . . 4 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑆 = ∅)
65fveq2d 6152 . . 3 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (Base‘𝑆) = (Base‘∅))
7 elbasov.b . . 3 𝐵 = (Base‘𝑆)
8 base0 15833 . . 3 ∅ = (Base‘∅)
96, 7, 83eqtr4g 2680 . 2 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝐵 = ∅)
101, 9nsyl2 142 1 (𝐴𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891  dom cdm 5074  Rel wrel 5079  cfv 5847  (class class class)co 6604  Basecbs 15781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-slot 15785  df-base 15786
This theorem is referenced by:  strov2rcl  15843  psrelbas  19298  psraddcl  19302  psrmulcllem  19306  psrvscafval  19309  psrvscacl  19312  resspsradd  19335  resspsrmul  19336  cphsubrglem  22885  mdegcl  23733
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