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Theorem resinsn 49490
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 5991 . . 3 Rel (𝐹 ↾ (𝐴 ∩ {𝐵}))
2 reldm0 5904 . . 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 4161 . . . 4 ((𝐴 ∩ {𝐵}) ∩ dom 𝐹) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
5 dmres 5998 . . . 4 dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((𝐴 ∩ {𝐵}) ∩ dom 𝐹)
6 inass 4179 . . . 4 ((dom 𝐹𝐴) ∩ {𝐵}) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
74, 5, 63eqtr4i 2795 . . 3 dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((dom 𝐹𝐴) ∩ {𝐵})
87eqeq1i 2767 . 2 (dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ((dom 𝐹𝐴) ∩ {𝐵}) = ∅)
9 disjsn 4670 . 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 1560  wcel 2142  cin 3903  c0 4285  {csn 4582  dom cdm 5647  cres 5649  Rel wrel 5652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-dm 5657  df-res 5659
This theorem is referenced by:  tposres2  49498
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