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Metamath Proof Explorer

Theorem pmtrdifellem2

Description: Lemma 2 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 pmtrdifellem2 
|- ( Q e. T -> dom ( S \ _I ) = dom ( Q \ _I ) )

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 3 difeq1i 
 |-  ( S \ _I ) = ( ( ( pmTrsp ` N ) ` dom ( Q \ _I ) ) \ _I )
5 4 dmeqi 
 |-  dom ( S \ _I ) = dom ( ( ( pmTrsp ` N ) ` dom ( Q \ _I ) ) \ _I )
6 eqid 
 |-  ( pmTrsp ` ( N \ { K } ) ) = ( pmTrsp ` ( N \ { K } ) )
7 6 1 pmtrfb 
 |-  ( Q e. T <-> ( ( N \ { K } ) e. _V /\ Q : ( N \ { K } ) -1-1-onto-> ( N \ { K } ) /\ dom ( Q \ _I ) ~~ 2o ) )
8 difsnexi 
 |-  ( ( N \ { K } ) e. _V -> N e. _V )
9 f1of 
 |-  ( Q : ( N \ { K } ) -1-1-onto-> ( N \ { K } ) -> Q : ( N \ { K } ) --> ( N \ { K } ) )
10 fdm 
 |-  ( Q : ( N \ { K } ) --> ( N \ { K } ) -> dom Q = ( N \ { K } ) )
11 difssd 
 |-  ( dom Q = ( N \ { K } ) -> ( Q \ _I ) C_ Q )
12 dmss 
 |-  ( ( Q \ _I ) C_ Q -> dom ( Q \ _I ) C_ dom Q )
13 11 12 syl 
 |-  ( dom Q = ( N \ { K } ) -> dom ( Q \ _I ) C_ dom Q )
14 difssd 
 |-  ( dom Q = ( N \ { K } ) -> ( N \ { K } ) C_ N )
15 sseq1 
 |-  ( dom Q = ( N \ { K } ) -> ( dom Q C_ N <-> ( N \ { K } ) C_ N ) )
16 14 15 mpbird 
 |-  ( dom Q = ( N \ { K } ) -> dom Q C_ N )
17 13 16 sstrd 
 |-  ( dom Q = ( N \ { K } ) -> dom ( Q \ _I ) C_ N )
18 9 10 17 3syl 
 |-  ( Q : ( N \ { K } ) -1-1-onto-> ( N \ { K } ) -> dom ( Q \ _I ) C_ N )
19 id 
 |-  ( dom ( Q \ _I ) ~~ 2o -> dom ( Q \ _I ) ~~ 2o )
20 8 18 19 3anim123i 
 |-  ( ( ( N \ { K } ) e. _V /\ Q : ( N \ { K } ) -1-1-onto-> ( N \ { K } ) /\ dom ( Q \ _I ) ~~ 2o ) -> ( N e. _V /\ dom ( Q \ _I ) C_ N /\ dom ( Q \ _I ) ~~ 2o ) )
21 7 20 sylbi 
 |-  ( Q e. T -> ( N e. _V /\ dom ( Q \ _I ) C_ N /\ dom ( Q \ _I ) ~~ 2o ) )
22 eqid 
 |-  ( pmTrsp ` N ) = ( pmTrsp ` N )
23 22 pmtrmvd 
 |-  ( ( N e. _V /\ dom ( Q \ _I ) C_ N /\ dom ( Q \ _I ) ~~ 2o ) -> dom ( ( ( pmTrsp ` N ) ` dom ( Q \ _I ) ) \ _I ) = dom ( Q \ _I ) )
24 21 23 syl 
 |-  ( Q e. T -> dom ( ( ( pmTrsp ` N ) ` dom ( Q \ _I ) ) \ _I ) = dom ( Q \ _I ) )
25 5 24 eqtrid 
 |-  ( Q e. T -> dom ( S \ _I ) = dom ( Q \ _I ) )