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

Proof of Theorem fnunres2
StepHypRef Expression
1 uncom 4158 . . 3 (𝐹𝐺) = (𝐺𝐹)
21reseq1i 5993 . 2 ((𝐹𝐺) ↾ 𝐵) = ((𝐺𝐹) ↾ 𝐵)
3 ineqcom 4210 . . . 4 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
4 fnunres1 6680 . . . 4 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ (𝐵𝐴) = ∅) → ((𝐺𝐹) ↾ 𝐵) = 𝐺)
53, 4syl3an3b 1407 . . 3 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ (𝐴𝐵) = ∅) → ((𝐺𝐹) ↾ 𝐵) = 𝐺)
653com12 1124 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐺𝐹) ↾ 𝐵) = 𝐺)
72, 6eqtrid 2789 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
Colors of variables: wff setvar class
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
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