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 i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = G )

Proof of Theorem fresaunres2disj

Step	Hyp	Ref	Expression
1		resundir 5075	. . 3 |- ( ( F u. G ) |` B ) = ( ( F |` B ) u. ( G |` B ) )
2		simp3 1030	. . . . 5 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( A i^i B ) = (/) )
3		ffn 5531	. . . . . . 7 |- ( F : A --> C -> F Fn A )
4	3	3ad2ant1 1049	. . . . . 6 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> F Fn A )
5		fnresdisj 5491	. . . . . 6 |- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) )
6	4, 5	syl 14	. . . . 5 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) )
7	2, 6	mpbid 147	. . . 4 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i 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 i^i 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 i^i B ) = (/) ) -> ( G |` B ) = G )
12	7, 11	uneq12d 3384	. . 3 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F |` B ) u. ( G |` B ) ) = ( (/) u. G ) )
13	1, 12	eqtrid 2283	. 2 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = ( (/) u. G ) )
14		uncom 3373	. . 3 |- ( (/) u. G ) = ( G u. (/) )
15		un0 3556	. . 3 |- ( G u. (/) ) = G
16	14, 15	eqtri 2259	. 2 |- ( (/) u. G ) = G
17	13, 16	eqtrdi 2287	1 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = G )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   u. cun 3218   i^i 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