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

Proof of Theorem fnunres2
StepHypRef Expression
1 uncom 4168 . . 3 (𝐹𝐺) = (𝐺𝐹)
21reseq1i 5996 . 2 ((𝐹𝐺) ↾ 𝐵) = ((𝐺𝐹) ↾ 𝐵)
3 ineqcom 4218 . . . 4 ((𝐴𝐵) = ∅ ↔ (𝐵𝐴) = ∅)
4 fnunres1 6681 . . . 4 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ (𝐵𝐴) = ∅) → ((𝐺𝐹) ↾ 𝐵) = 𝐺)
53, 4syl3an3b 1404 . . 3 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ (𝐴𝐵) = ∅) → ((𝐺𝐹) ↾ 𝐵) = 𝐺)
653com12 1122 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐺𝐹) ↾ 𝐵) = 𝐺)
72, 6eqtrid 2787 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  cun 3961  cin 3962  c0 4339  cres 5691   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-res 5701  df-fun 6565  df-fn 6566
This theorem is referenced by:  tocycfvres1  33113  cycpmconjslem2  33158  dvun  42368  evlselvlem  42573  evlselv  42574
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