Theorem fnunres2 6606

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 4099	. . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
2	1	reseq1i 5935	. 2 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐵) = ((𝐺 ∪ 𝐹) ↾ 𝐵)
3		ineqcom 4151	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)
4		fnunres1 6605	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ (𝐵 ∩ 𝐴) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺)
5	3, 4	syl3an3b 1408	. . 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 2784	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∪ cun 3888   ∩ cin 3889  ∅c0 4274   ↾ cres 5627   Fn wfn 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-dm 5635  df-res 5637  df-fun 6495  df-fn 6496

This theorem is referenced by:  tocycfvres1  33189  cycpmconjslem2  33234  dvun  42808  evlselvlem  43036  evlselv  43037