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Theorem fnunres2 30735
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 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)

Proof of Theorem fnunres2
StepHypRef Expression
1 uncom 4067 . . 3 (𝐹𝐺) = (𝐺𝐹)
21reseq1i 5847 . 2 ((𝐹𝐺) ↾ 𝐵) = ((𝐺𝐹) ↾ 𝐵)
3 simp2 1139 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → 𝐺 Fn 𝐵)
4 simp1 1138 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → 𝐹 Fn 𝐴)
5 incom 4115 . . . 4 (𝐴𝐵) = (𝐵𝐴)
6 simp3 1140 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
75, 6eqtr3id 2792 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (𝐵𝐴) = ∅)
8 fnunres1 30664 . . 3 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ (𝐵𝐴) = ∅) → ((𝐺𝐹) ↾ 𝐵) = 𝐺)
93, 4, 7, 8syl3anc 1373 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐺𝐹) ↾ 𝐵) = 𝐺)
102, 9syl5eq 2790 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  cun 3864  cin 3865  c0 4237  cres 5553   Fn wfn 6375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-dm 5561  df-res 5563  df-fun 6382  df-fn 6383
This theorem is referenced by:  tocycfvres1  31096  cycpmconjslem2  31141
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