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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 )