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Theorem fnunres1 6681
Description: Restriction of a disjoint union to the domain of the first function. (Contributed by Thierry Arnoux, 9-Dec-2021.)
Assertion
Ref Expression
fnunres1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)

Proof of Theorem fnunres1
StepHypRef Expression
1 fndm 6672 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
213ad2ant1 1132 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → dom 𝐹 = 𝐴)
32reseq2d 6000 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ dom 𝐹) = ((𝐹𝐺) ↾ 𝐴))
4 fnrel 6671 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
543ad2ant1 1132 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → Rel 𝐹)
6 fndm 6672 . . . . . 6 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
763ad2ant2 1133 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → dom 𝐺 = 𝐵)
82, 7ineq12d 4229 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
9 simp3 1137 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
108, 9eqtrd 2775 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = ∅)
11 relresdm1 6053 . . 3 ((Rel 𝐹 ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺) ↾ dom 𝐹) = 𝐹)
125, 10, 11syl2anc 584 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ dom 𝐹) = 𝐹)
133, 12eqtr3d 2777 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  cun 3961  cin 3962  c0 4339  dom cdm 5689  cres 5691  Rel wrel 5694   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:  fnunres2  6682  tocycfvres2  33114  cycpmconjslem2  33158  lbsdiflsp0  33654  actfunsnf1o  34598  dvun  42368  evlselvlem  42573  evlselv  42574
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