Theorem fresaunres2disj 5547

Description: From the union of two functions with disjoint domains, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Jim Kingdon, 18-May-2026.)

Assertion

Ref	Expression
fresaunres2disj	⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)

Proof of Theorem fresaunres2disj

Step	Hyp	Ref	Expression
1		resundir 5054	. . 3 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐵) = ((𝐹 ↾ 𝐵) ∪ (𝐺 ↾ 𝐵))
2		simp3 1026	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
3		ffn 5510	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐶 → 𝐹 Fn 𝐴)
4	3	3ad2ant1 1045	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 Fn 𝐴)
5		fnresdisj 5470	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐹 ↾ 𝐵) = ∅))
6	4, 5	syl 14	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐹 ↾ 𝐵) = ∅))
7	2, 6	mpbid 147	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ↾ 𝐵) = ∅)
8		ffn 5510	. . . . . 6 ⊢ (𝐺:𝐵⟶𝐶 → 𝐺 Fn 𝐵)
9	8	3ad2ant2 1046	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐺 Fn 𝐵)
10		fnresdm 5469	. . . . 5 ⊢ (𝐺 Fn 𝐵 → (𝐺 ↾ 𝐵) = 𝐺)
11	9, 10	syl 14	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐺 ↾ 𝐵) = 𝐺)
12	7, 11	uneq12d 3376	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ↾ 𝐵) ∪ (𝐺 ↾ 𝐵)) = (∅ ∪ 𝐺))
13	1, 12	eqtrid 2279	. 2 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = (∅ ∪ 𝐺))
14		uncom 3365	. . 3 ⊢ (∅ ∪ 𝐺) = (𝐺 ∪ ∅)
15		un0 3544	. . 3 ⊢ (𝐺 ∪ ∅) = 𝐺
16	14, 15	eqtri 2255	. 2 ⊢ (∅ ∪ 𝐺) = 𝐺
17	13, 16	eqtrdi 2283	1 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∪ cun 3211   ∩ cin 3212  ∅c0 3510   ↾ cres 4753   Fn wfn 5349  ⟶wf 5350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-dm 4761  df-res 4763  df-fun 5356  df-fn 5357  df-f 5358

This theorem is referenced by:  fresaunres1disj  5548  mapunen  7106