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Theorem fnunres1 30284
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 6448 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
213ad2ant1 1125 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → dom 𝐹 = 𝐴)
32reseq2d 5846 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ dom 𝐹) = ((𝐹𝐺) ↾ 𝐴))
4 fnrel 6447 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
543ad2ant1 1125 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → Rel 𝐹)
6 fndm 6448 . . . . . 6 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
763ad2ant2 1126 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → dom 𝐺 = 𝐵)
82, 7ineq12d 4187 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
9 simp3 1130 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
108, 9eqtrd 2853 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = ∅)
11 funresdm1 30283 . . 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 2855 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  cun 3931  cin 3932  c0 4288  dom cdm 5548  cres 5550  Rel wrel 5553   Fn wfn 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-dm 5558  df-res 5560  df-fun 6350  df-fn 6351
This theorem is referenced by:  fnunres2  30352  tocycfvres2  30680  cycpmconjslem2  30724  lbsdiflsp0  30921  actfunsnf1o  31774
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