Metamath Proof Explorer


Theorem psrbaglecl

Description: The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024)

Ref Expression
Hypothesis psrbag.d
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
Assertion psrbaglecl
|- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> G e. D )

Proof

Step Hyp Ref Expression
1 psrbag.d
 |-  D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
2 simp2
 |-  ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> G : I --> NN0 )
3 simp1
 |-  ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> F e. D )
4 id
 |-  ( F e. D -> F e. D )
5 1 psrbagf
 |-  ( F e. D -> F : I --> NN0 )
6 5 ffnd
 |-  ( F e. D -> F Fn I )
7 4 6 fndmexd
 |-  ( F e. D -> I e. _V )
8 7 3ad2ant1
 |-  ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> I e. _V )
9 1 psrbag
 |-  ( I e. _V -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) )
10 8 9 syl
 |-  ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) )
11 3 10 mpbid
 |-  ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) )
12 11 simprd
 |-  ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( `' F " NN ) e. Fin )
13 1 psrbaglesupp
 |-  ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( `' G " NN ) C_ ( `' F " NN ) )
14 12 13 ssfid
 |-  ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( `' G " NN ) e. Fin )
15 1 psrbag
 |-  ( I e. _V -> ( G e. D <-> ( G : I --> NN0 /\ ( `' G " NN ) e. Fin ) ) )
16 8 15 syl
 |-  ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( G e. D <-> ( G : I --> NN0 /\ ( `' G " NN ) e. Fin ) ) )
17 2 14 16 mpbir2and
 |-  ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> G e. D )