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

Proof of Theorem fresaunres2disj

Step	Hyp	Ref	Expression
1		resundir 5054	. . 3 |- ( ( F u. G ) |` B ) = ( ( F |` B ) u. ( G |` B ) )
2		simp3 1026	. . . . 5 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( A i^i B ) = (/) )
3		ffn 5510	. . . . . . 7 |- ( F : A --> C -> F Fn A )
4	3	3ad2ant1 1045	. . . . . 6 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> F Fn A )
5		fnresdisj 5470	. . . . . 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 5510	. . . . . 6 |- ( G : B --> C -> G Fn B )
9	8	3ad2ant2 1046	. . . . 5 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i 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 i^i B ) = (/) ) -> ( G |` B ) = G )
12	7, 11	uneq12d 3376	. . 3 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F |` B ) u. ( G |` B ) ) = ( (/) u. G ) )
13	1, 12	eqtrid 2279	. 2 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = ( (/) u. G ) )
14		uncom 3365	. . 3 |- ( (/) u. G ) = ( G u. (/) )
15		un0 3544	. . 3 |- ( G u. (/) ) = G
16	14, 15	eqtri 2255	. 2 |- ( (/) u. G ) = G
17	13, 16	eqtrdi 2283	1 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = G )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   = wceq 1398   u. cun 3211   i^i 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