Theorem relresdm1 5988

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 5949	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴))
2		resdm 5981	. . . . 5 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
3	2	adantr 481	. . . 4 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐴 ↾ dom 𝐴) = 𝐴)
4		dmres 5967	. . . . . 6 ⊢ dom (𝐵 ↾ dom 𝐴) = (dom 𝐴 ∩ dom 𝐵)
5		simpr 485	. . . . . 6 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (dom 𝐴 ∩ dom 𝐵) = ∅)
6	4, 5	eqtrid 2783	. . . . 5 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → dom (𝐵 ↾ dom 𝐴) = ∅)
7		relres 5960	. . . . . 6 ⊢ Rel (𝐵 ↾ dom 𝐴)
8		reldm0 5873	. . . . . 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 235	. . . 4 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → (𝐵 ↾ dom 𝐴) = ∅)
11	3, 10	uneq12d 4102	. . 3 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = (𝐴 ∪ ∅))
12		un0 4325	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
13	11, 12	eqtrdi 2787	. 2 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ↾ dom 𝐴) ∪ (𝐵 ↾ dom 𝐴)) = 𝐴)
14	1, 13	eqtrid 2783	1 ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1543   ∪ cun 3884   ∩ cin 3885  ∅c0 4264  dom cdm 5621   ↾ cres 5623  Rel wrel 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-ext 2708  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-sb 2070  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3051  df-rex 3061  df-rab 3389  df-v 3430  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-dm 5631  df-res 5633

This theorem is referenced by:  fnunres1  6600