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Metamath Proof Explorer

Theorem eqresfnbd

Description: Property of being the restriction of a function. Note that this is closer to funssres than fnssres . (Contributed by SN, 11-Mar-2025)

Ref Expression
Hypotheses eqresfnbd.g 
|- ( ph -> F Fn B )
eqresfnbd.1 
|- ( ph -> A C_ B )
Assertion eqresfnbd 
|- ( ph -> ( R = ( F |` A ) <-> ( R Fn A /\ R C_ F ) ) )

Proof

Step Hyp Ref Expression
1 eqresfnbd.g 
 |-  ( ph -> F Fn B )
2 eqresfnbd.1 
 |-  ( ph -> A C_ B )
3 1 2 fnssresd 
 |-  ( ph -> ( F |` A ) Fn A )
4 resss 
 |-  ( F |` A ) C_ F
5 3 4 jctir 
 |-  ( ph -> ( ( F |` A ) Fn A /\ ( F |` A ) C_ F ) )
6 fneq1 
 |-  ( R = ( F |` A ) -> ( R Fn A <-> ( F |` A ) Fn A ) )
7 sseq1 
 |-  ( R = ( F |` A ) -> ( R C_ F <-> ( F |` A ) C_ F ) )
8 6 7 anbi12d 
 |-  ( R = ( F |` A ) -> ( ( R Fn A /\ R C_ F ) <-> ( ( F |` A ) Fn A /\ ( F |` A ) C_ F ) ) )
9 5 8 syl5ibrcom 
 |-  ( ph -> ( R = ( F |` A ) -> ( R Fn A /\ R C_ F ) ) )
10 1 fnfund 
 |-  ( ph -> Fun F )
11 10 adantr 
 |-  ( ( ph /\ R Fn A ) -> Fun F )
12 funssres 
 |-  ( ( Fun F /\ R C_ F ) -> ( F |` dom R ) = R )
13 12 eqcomd 
 |-  ( ( Fun F /\ R C_ F ) -> R = ( F |` dom R ) )
14 fndm 
 |-  ( R Fn A -> dom R = A )
15 14 adantl 
 |-  ( ( ph /\ R Fn A ) -> dom R = A )
16 15 reseq2d 
 |-  ( ( ph /\ R Fn A ) -> ( F |` dom R ) = ( F |` A ) )
17 16 eqeq2d 
 |-  ( ( ph /\ R Fn A ) -> ( R = ( F |` dom R ) <-> R = ( F |` A ) ) )
18 13 17 imbitrid 
 |-  ( ( ph /\ R Fn A ) -> ( ( Fun F /\ R C_ F ) -> R = ( F |` A ) ) )
19 11 18 mpand 
 |-  ( ( ph /\ R Fn A ) -> ( R C_ F -> R = ( F |` A ) ) )
20 19 expimpd 
 |-  ( ph -> ( ( R Fn A /\ R C_ F ) -> R = ( F |` A ) ) )
21 9 20 impbid 
 |-  ( ph -> ( R = ( F |` A ) <-> ( R Fn A /\ R C_ F ) ) )