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Theorem funresdm1 30283
Description: Restriction of a disjoint union to the domain of the first term. (Contributed by Thierry Arnoux, 9-Dec-2021.)
Assertion
Ref Expression
funresdm1 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴𝐵) ↾ dom 𝐴) = 𝐴)

Proof of Theorem funresdm1
StepHypRef Expression
1 resundir 5861 . 2 ((𝐴𝐵) ↾ dom 𝐴) = ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴))
2 resdm 5890 . . . . 5 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
32adantr 481 . . . 4 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐴 ↾ dom 𝐴) = 𝐴)
4 dmres 5868 . . . . . 6 dom (𝐵 ↾ dom 𝐴) = (dom 𝐴 ∩ dom 𝐵)
5 simpr 485 . . . . . 6 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (dom 𝐴 ∩ dom 𝐵) = ∅)
64, 5syl5eq 2865 . . . . 5 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → dom (𝐵 ↾ dom 𝐴) = ∅)
7 relres 5875 . . . . . 6 Rel (𝐵 ↾ dom 𝐴)
8 reldm0 5791 . . . . . 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 235 . . . 4 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐵 ↾ dom 𝐴) = ∅)
113, 10uneq12d 4137 . . 3 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = (𝐴 ∪ ∅))
12 un0 4341 . . 3 (𝐴 ∪ ∅) = 𝐴
1311, 12syl6eq 2869 . 2 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = 𝐴)
141, 13syl5eq 2865 1 ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴𝐵) ↾ dom 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  cun 3931  cin 3932  c0 4288  dom cdm 5548  cres 5550  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-dm 5558  df-res 5560
This theorem is referenced by:  fnunres1  30284
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