Theorem fresaunres1disj 5569

Description: From the union of two functions with disjoint domains, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.) (Revised by Jim Kingdon, 18-May-2026.)

Assertion

Ref	Expression
fresaunres1disj	|- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` A ) = F )

Proof of Theorem fresaunres1disj

Step	Hyp	Ref	Expression
1		fresaunres2disj 5568	. . 3 |- ( ( G : B --> C /\ F : A --> C /\ ( B i^i A ) = (/) ) -> ( ( G u. F ) |` A ) = F )
2	1	3com12 1238	. 2 |- ( ( F : A --> C /\ G : B --> C /\ ( B i^i A ) = (/) ) -> ( ( G u. F ) |` A ) = F )
3		incom 3421	. . . 4 |- ( B i^i A ) = ( A i^i B )
4	3	eqeq1i 2246	. . 3 |- ( ( B i^i A ) = (/) <-> ( A i^i B ) = (/) )
5	4	3anbi3i 1223	. 2 |- ( ( F : A --> C /\ G : B --> C /\ ( B i^i A ) = (/) ) <-> ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) )
6		uncom 3373	. . . 4 |- ( G u. F ) = ( F u. G )
7	6	reseq1i 5057	. . 3 |- ( ( G u. F ) |` A ) = ( ( F u. G ) |` A )
8	7	eqeq1i 2246	. 2 |- ( ( ( G u. F ) |` A ) = F <-> ( ( F u. G ) |` A ) = F )
9	2, 5, 8	3imtr3i 200	1 |- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` A ) = F )

Syntax hints:   -> wi 4   /\ w3a 1009   = wceq 1402   u. cun 3218   i^i cin 3219   (/)c0 3520   |` cres 4774   -->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:  mapunen  7144  hashf1lem1  11266