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Theorem fnunres2 30485
 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 4083 . . 3 (𝐹𝐺) = (𝐺𝐹)
21reseq1i 5818 . 2 ((𝐹𝐺) ↾ 𝐵) = ((𝐺𝐹) ↾ 𝐵)
3 simp2 1134 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → 𝐺 Fn 𝐵)
4 simp1 1133 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → 𝐹 Fn 𝐴)
5 incom 4131 . . . 4 (𝐴𝐵) = (𝐵𝐴)
6 simp3 1135 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
75, 6syl5eqr 2847 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → (𝐵𝐴) = ∅)
8 fnunres1 30413 . . 3 ((𝐺 Fn 𝐵𝐹 Fn 𝐴 ∧ (𝐵𝐴) = ∅) → ((𝐺𝐹) ↾ 𝐵) = 𝐺)
93, 4, 7, 8syl3anc 1368 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐺𝐹) ↾ 𝐵) = 𝐺)
102, 9syl5eq 2845 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∪ cun 3881   ∩ cin 3882  ∅c0 4246   ↾ cres 5525   Fn wfn 6327 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-xp 5529  df-rel 5530  df-dm 5533  df-res 5535  df-fun 6334  df-fn 6335 This theorem is referenced by:  tocycfvres1  30851  cycpmconjslem2  30896
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