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Theorem elbasov 16536
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 4271 . 2 (𝐴𝐵 → ¬ 𝐵 = ∅)
2 elbasov.s . . . . 5 𝑆 = (𝑋𝑂𝑌)
3 elbasov.o . . . . . 6 Rel dom 𝑂
43ovprc 7178 . . . . 5 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅)
52, 4syl5eq 2869 . . . 4 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑆 = ∅)
65fveq2d 6656 . . 3 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (Base‘𝑆) = (Base‘∅))
7 elbasov.b . . 3 𝐵 = (Base‘𝑆)
8 base0 16527 . . 3 ∅ = (Base‘∅)
96, 7, 83eqtr4g 2882 . 2 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝐵 = ∅)
101, 9nsyl2 143 1 (𝐴𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2114  Vcvv 3469  c0 4265  dom cdm 5532  Rel wrel 5537  cfv 6334  (class class class)co 7140  Basecbs 16474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-slot 16478  df-base 16480
This theorem is referenced by:  strov2rcl  16537  psrelbas  20615  psraddcl  20619  psrmulcllem  20623  psrvscafval  20626  psrvscacl  20629  resspsradd  20652  resspsrmul  20653  cphsubrglem  23780  mdegcl  24668
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