Theorem fnunres2 6682

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 4160	. . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
2	1	reseq1i 5982	. 2 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐵) = ((𝐺 ∪ 𝐹) ↾ 𝐵)
3		ineqcom 4110	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)
4		fnunres1 6681	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ (𝐵 ∩ 𝐴) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺)
5	3, 4	syl3an3b 1433	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺)
6	5	3com12 1141	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺)
7	2, 6	eqtrid 2810	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∩ cin 4092   ∪ cun 4152  ∅c0 4281   ↾ cres 5660   Fn wfn 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-dm 5668  df-res 5670  df-fun 6568  df-fn 6569

This theorem is used by:  tocycfvres1  33770  cycpmconjslem2  33815  dvun  43676  evlselvlem  43882  evlselv  43883