Theorem fresaunres2disj 5568

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 5075	. . 3 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐵) = ((𝐹 ↾ 𝐵) ∪ (𝐺 ↾ 𝐵))
2		simp3 1030	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
3		ffn 5531	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐶 → 𝐹 Fn 𝐴)
4	3	3ad2ant1 1049	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 Fn 𝐴)
5		fnresdisj 5491	. . . . . 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 5531	. . . . . 6 ⊢ (𝐺:𝐵⟶𝐶 → 𝐺 Fn 𝐵)
9	8	3ad2ant2 1050	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐺 Fn 𝐵)
10		fnresdm 5490	. . . . 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 3384	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ↾ 𝐵) ∪ (𝐺 ↾ 𝐵)) = (∅ ∪ 𝐺))
13	1, 12	eqtrid 2283	. 2 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = (∅ ∪ 𝐺))
14		uncom 3373	. . 3 ⊢ (∅ ∪ 𝐺) = (𝐺 ∪ ∅)
15		un0 3556	. . 3 ⊢ (𝐺 ∪ ∅) = 𝐺
16	14, 15	eqtri 2259	. 2 ⊢ (∅ ∪ 𝐺) = 𝐺
17	13, 16	eqtrdi 2287	1 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∪ cun 3218   ∩ cin 3219  ∅c0 3520   ↾ cres 4774   Fn wfn 5370  ⟶wf 5371

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-dm 4782  df-res 4784  df-fun 5377  df-fn 5378  df-f 5379

This theorem is referenced by:  fresaunres1disj  5569  mapunen  7144