Theorem fnunres1 6630

Description: Restriction of a disjoint union to the domain of the first function. (Contributed by Thierry Arnoux, 9-Dec-2021.)

Assertion

Ref	Expression
fnunres1	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)

Proof of Theorem fnunres1

Step	Hyp	Ref	Expression
1		fndm 6621	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2	1	3ad2ant1 1133	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → dom 𝐹 = 𝐴)
3	2	reseq2d 5950	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = ((𝐹 ∪ 𝐺) ↾ 𝐴))
4		fnrel 6620	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
5	4	3ad2ant1 1133	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → Rel 𝐹)
6		fndm 6621	. . . . . 6 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
7	6	3ad2ant2 1134	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → dom 𝐺 = 𝐵)
8	2, 7	ineq12d 4184	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
9		simp3 1138	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
10	8, 9	eqtrd 2764	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = ∅)
11		relresdm1 6004	. . 3 ⊢ ((Rel 𝐹 ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = 𝐹)
12	5, 10, 11	syl2anc 584	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = 𝐹)
13	3, 12	eqtr3d 2766	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∪ cun 3912   ∩ cin 3913  ∅c0 4296  dom cdm 5638   ↾ cres 5640  Rel wrel 5643   Fn wfn 6506

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dm 5648  df-res 5650  df-fun 6513  df-fn 6514

This theorem is referenced by:  fnunres2  6631  tocycfvres2  33068  cycpmconjslem2  33112  lbsdiflsp0  33622  actfunsnf1o  34595  dvun  42347  evlselvlem  42574  evlselv  42575