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Theorem relresdm1 6051
Description: Restriction of a disjoint union to the domain of the first term. (Contributed by Thierry Arnoux, 9-Dec-2021.)
Assertion
Ref Expression
relresdm1 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴𝐵) ↾ dom 𝐴) = 𝐴)

Proof of Theorem relresdm1
StepHypRef Expression
1 resundir 6012 . 2 ((𝐴𝐵) ↾ dom 𝐴) = ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴))
2 resdm 6044 . . . . 5 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
32adantr 480 . . . 4 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐴 ↾ dom 𝐴) = 𝐴)
4 dmres 6030 . . . . . 6 dom (𝐵 ↾ dom 𝐴) = (dom 𝐴 ∩ dom 𝐵)
5 simpr 484 . . . . . 6 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (dom 𝐴 ∩ dom 𝐵) = ∅)
64, 5eqtrid 2789 . . . . 5 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → dom (𝐵 ↾ dom 𝐴) = ∅)
7 relres 6023 . . . . . 6 Rel (𝐵 ↾ dom 𝐴)
8 reldm0 5938 . . . . . 6 (Rel (𝐵 ↾ dom 𝐴) → ((𝐵 ↾ dom 𝐴) = ∅ ↔ dom (𝐵 ↾ dom 𝐴) = ∅))
97, 8ax-mp 5 . . . . 5 ((𝐵 ↾ dom 𝐴) = ∅ ↔ dom (𝐵 ↾ dom 𝐴) = ∅)
106, 9sylibr 234 . . . 4 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐵 ↾ dom 𝐴) = ∅)
113, 10uneq12d 4169 . . 3 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = (𝐴 ∪ ∅))
12 un0 4394 . . 3 (𝐴 ∪ ∅) = 𝐴
1311, 12eqtrdi 2793 . 2 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = 𝐴)
141, 13eqtrid 2789 1 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴𝐵) ↾ dom 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  cun 3949  cin 3950  c0 4333  dom cdm 5685  cres 5687  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dm 5695  df-res 5697
This theorem is referenced by:  fnunres1  6680
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