Theorem relresdm1 6024

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) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴)

Proof of Theorem relresdm1

Step	Hyp	Ref	Expression
1		resundir 5984	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴))
2		resdm 6014	. . . . 5 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
3	2	adantr 486	. . . 4 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐴 ↾ dom 𝐴) = 𝐴)
4		dmres 6002	. . . . . 6 ⊢ dom (𝐵 ↾ dom 𝐴) = (dom 𝐴 ∩ dom 𝐵)
5		simpr 490	. . . . . 6 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (dom 𝐴 ∩ dom 𝐵) = ∅)
6	4, 5	eqtrid 2807	. . . . 5 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → dom (𝐵 ↾ dom 𝐴) = ∅)
7		relres 5995	. . . . . 6 ⊢ Rel (𝐵 ↾ dom 𝐴)
8		reldm0 5909	. . . . . 6 ⊢ (Rel (𝐵 ↾ dom 𝐴) → ((𝐵 ↾ dom 𝐴) = ∅ ↔ dom (𝐵 ↾ dom 𝐴) = ∅))
9	7, 8	ax-mp 5	. . . . 5 ⊢ ((𝐵 ↾ dom 𝐴) = ∅ ↔ dom (𝐵 ↾ dom 𝐴) = ∅)
10	6, 9	sylibr 237	. . . 4 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐵 ↾ dom 𝐴) = ∅)
11	3, 10	uneq12d 4168	. . 3 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = (𝐴 ∪ ∅))
12		un0 4343	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
13	11, 12	eqtrdi 2811	. 2 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = 𝐴)
14	1, 13	eqtrid 2807	1 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  dom cdm 5650   ↾ cres 5652  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dm 5660  df-res 5662

This theorem is used by:  fnunres1  6652