Theorem fnunres2 6681

Description: Restriction of a disjoint union to the domain of the second function. (Contributed by Thierry Arnoux, 12-Oct-2023.)

Assertion

Ref	Expression
fnunres2	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)

Proof of Theorem fnunres2

Step	Hyp	Ref	Expression
1		uncom 4158	. . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
2	1	reseq1i 5993	. 2 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐵) = ((𝐺 ∪ 𝐹) ↾ 𝐵)
3		ineqcom 4210	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)
4		fnunres1 6680	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ (𝐵 ∩ 𝐴) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺)
5	3, 4	syl3an3b 1407	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺)
6	5	3com12 1124	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺)
7	2, 6	eqtrid 2789	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1540   ∪ cun 3949   ∩ cin 3950  ∅c0 4333   ↾ cres 5687   Fn wfn 6556

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  df-fun 6563  df-fn 6564

This theorem is referenced by:  tocycfvres1  33130  cycpmconjslem2  33175  dvun  42389  evlselvlem  42596  evlselv  42597