Metamath Proof Explorer

Theorem pmtrdifellem1

Description: Lemma 1 for pmtrdifel . (Contributed by AV, 15-Jan-2019)

Ref Expression
Hypotheses pmtrdifel.t 
|- T = ran ( pmTrsp ` ( N \ { K } ) )
pmtrdifel.r 
|- R = ran ( pmTrsp ` N )
pmtrdifel.0 
|- S = ( ( pmTrsp ` N ) ` dom ( Q \ _I ) )
Assertion pmtrdifellem1 
|- ( Q e. T -> S e. R )

Proof

Step Hyp Ref Expression
1 pmtrdifel.t 
 |-  T = ran ( pmTrsp ` ( N \ { K } ) )
2 pmtrdifel.r 
 |-  R = ran ( pmTrsp ` N )
3 pmtrdifel.0 
 |-  S = ( ( pmTrsp ` N ) ` dom ( Q \ _I ) )
4 eqid 
 |-  ( pmTrsp ` ( N \ { K } ) ) = ( pmTrsp ` ( N \ { K } ) )
5 4 1 pmtrfb 
 |-  ( Q e. T <-> ( ( N \ { K } ) e. _V /\ Q : ( N \ { K } ) -1-1-onto-> ( N \ { K } ) /\ dom ( Q \ _I ) ~~ 2o ) )
6 difsnexi 
 |-  ( ( N \ { K } ) e. _V -> N e. _V )
7 f1of 
 |-  ( Q : ( N \ { K } ) -1-1-onto-> ( N \ { K } ) -> Q : ( N \ { K } ) --> ( N \ { K } ) )
8 fdm 
 |-  ( Q : ( N \ { K } ) --> ( N \ { K } ) -> dom Q = ( N \ { K } ) )
9 difssd 
 |-  ( dom Q = ( N \ { K } ) -> ( Q \ _I ) C_ Q )
10 dmss 
 |-  ( ( Q \ _I ) C_ Q -> dom ( Q \ _I ) C_ dom Q )
11 9 10 syl 
 |-  ( dom Q = ( N \ { K } ) -> dom ( Q \ _I ) C_ dom Q )
12 difssd 
 |-  ( dom Q = ( N \ { K } ) -> ( N \ { K } ) C_ N )
13 sseq1 
 |-  ( dom Q = ( N \ { K } ) -> ( dom Q C_ N <-> ( N \ { K } ) C_ N ) )
14 12 13 mpbird 
 |-  ( dom Q = ( N \ { K } ) -> dom Q C_ N )
15 11 14 sstrd 
 |-  ( dom Q = ( N \ { K } ) -> dom ( Q \ _I ) C_ N )
16 7 8 15 3syl 
 |-  ( Q : ( N \ { K } ) -1-1-onto-> ( N \ { K } ) -> dom ( Q \ _I ) C_ N )
17 id 
 |-  ( dom ( Q \ _I ) ~~ 2o -> dom ( Q \ _I ) ~~ 2o )
18 eqid 
 |-  ( pmTrsp ` N ) = ( pmTrsp ` N )
19 18 2 pmtrrn 
 |-  ( ( N e. _V /\ dom ( Q \ _I ) C_ N /\ dom ( Q \ _I ) ~~ 2o ) -> ( ( pmTrsp ` N ) ` dom ( Q \ _I ) ) e. R )
20 3 19 eqeltrid 
 |-  ( ( N e. _V /\ dom ( Q \ _I ) C_ N /\ dom ( Q \ _I ) ~~ 2o ) -> S e. R )
21 6 16 17 20 syl3an 
 |-  ( ( ( N \ { K } ) e. _V /\ Q : ( N \ { K } ) -1-1-onto-> ( N \ { K } ) /\ dom ( Q \ _I ) ~~ 2o ) -> S e. R )
22 5 21 sylbi 
 |-  ( Q e. T -> S e. R )