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Theorem fnunres1 6655
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 6646 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
213ad2ant1 1130 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → dom 𝐹 = 𝐴)
32reseq2d 5975 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ dom 𝐹) = ((𝐹𝐺) ↾ 𝐴))
4 fnrel 6645 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
543ad2ant1 1130 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → Rel 𝐹)
6 fndm 6646 . . . . . 6 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
763ad2ant2 1131 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → dom 𝐺 = 𝐵)
82, 7ineq12d 4208 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
9 simp3 1135 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
108, 9eqtrd 2766 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = ∅)
11 relresdm1 6027 . . 3 ((Rel 𝐹 ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺) ↾ dom 𝐹) = 𝐹)
125, 10, 11syl2anc 583 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ dom 𝐹) = 𝐹)
133, 12eqtr3d 2768 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  cun 3941  cin 3942  c0 4317  dom cdm 5669  cres 5671  Rel wrel 5674   Fn wfn 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-dm 5679  df-res 5681  df-fun 6539  df-fn 6540
This theorem is referenced by:  fnunres2  6656  tocycfvres2  32776  cycpmconjslem2  32820  lbsdiflsp0  33229  actfunsnf1o  34145  evlselvlem  41712  evlselv  41713
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