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Theorem fnunres1 29724
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 6151 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
213ad2ant1 1128 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → dom 𝐹 = 𝐴)
32reseq2d 5551 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ dom 𝐹) = ((𝐹𝐺) ↾ 𝐴))
4 fnrel 6150 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
543ad2ant1 1128 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → Rel 𝐹)
6 fndm 6151 . . . . . 6 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
763ad2ant2 1129 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → dom 𝐺 = 𝐵)
82, 7ineq12d 3958 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
9 simp3 1133 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
108, 9eqtrd 2794 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = ∅)
11 funresdm1 29723 . . 3 ((Rel 𝐹 ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺) ↾ dom 𝐹) = 𝐹)
125, 10, 11syl2anc 696 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ dom 𝐹) = 𝐹)
133, 12eqtr3d 2796 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐴) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  cun 3713  cin 3714  c0 4058  dom cdm 5266  cres 5268  Rel wrel 5271   Fn wfn 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-dm 5276  df-res 5278  df-fun 6051  df-fn 6052
This theorem is referenced by:  actfunsnf1o  30991
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