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Theorem resinsn 49454
Description: Restriction to the intersection with a singleton. (Contributed by Zhi Wang, 6-Oct-2025.)
Assertion
Ref Expression
resinsn ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹𝐴))

Proof of Theorem resinsn
StepHypRef Expression
1 relres 5987 . . 3 Rel (𝐹 ↾ (𝐴 ∩ {𝐵}))
2 reldm0 5900 . . 3 (Rel (𝐹 ↾ (𝐴 ∩ {𝐵})) → ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅))
31, 2ax-mp 5 . 2 ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅)
4 incom 4159 . . . 4 ((𝐴 ∩ {𝐵}) ∩ dom 𝐹) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
5 dmres 5994 . . . 4 dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((𝐴 ∩ {𝐵}) ∩ dom 𝐹)
6 inass 4177 . . . 4 ((dom 𝐹𝐴) ∩ {𝐵}) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
74, 5, 63eqtr4i 2794 . . 3 dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((dom 𝐹𝐴) ∩ {𝐵})
87eqeq1i 2766 . 2 (dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ((dom 𝐹𝐴) ∩ {𝐵}) = ∅)
9 disjsn 4667 . 2 (((dom 𝐹𝐴) ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹𝐴))
103, 8, 93bitri 299 1 ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208   = wceq 1559  wcel 2141  cin 3901  c0 4283  {csn 4579  dom cdm 5643  cres 5645  Rel wrel 5648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-rel 5650  df-dm 5653  df-res 5655
This theorem is referenced by:  tposres2  49462
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