Metamath Proof Explorer

Theorem fparlem4

Description: Lemma for fpar . (Contributed by NM, 22-Dec-2008) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion fparlem4 
|- ( G Fn B -> ( `' ( 2nd |` ( _V X. _V ) ) o. ( G o. ( 2nd |` ( _V X. _V ) ) ) ) = U_ y e. B ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) ) )

Proof

Step Hyp Ref Expression
1 coiun 
 |-  ( `' ( 2nd |` ( _V X. _V ) ) o. U_ y e. B ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) ) = U_ y e. B ( `' ( 2nd |` ( _V X. _V ) ) o. ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) )
2 inss1 
 |-  ( dom G i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ dom G
3 fndm 
 |-  ( G Fn B -> dom G = B )
4 2 3 sseqtrid 
 |-  ( G Fn B -> ( dom G i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ B )
5 dfco2a 
 |-  ( ( dom G i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ B -> ( G o. ( 2nd |` ( _V X. _V ) ) ) = U_ y e. B ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) )
6 4 5 syl 
 |-  ( G Fn B -> ( G o. ( 2nd |` ( _V X. _V ) ) ) = U_ y e. B ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) )
7 6 coeq2d 
 |-  ( G Fn B -> ( `' ( 2nd |` ( _V X. _V ) ) o. ( G o. ( 2nd |` ( _V X. _V ) ) ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. U_ y e. B ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) ) )
8 inss1 
 |-  ( dom ( { ( G ` y ) } X. ( _V X. { y } ) ) i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ dom ( { ( G ` y ) } X. ( _V X. { y } ) )
9 dmxpss 
 |-  dom ( { ( G ` y ) } X. ( _V X. { y } ) ) C_ { ( G ` y ) }
10 8 9 sstri 
 |-  ( dom ( { ( G ` y ) } X. ( _V X. { y } ) ) i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ { ( G ` y ) }
11 dfco2a 
 |-  ( ( dom ( { ( G ` y ) } X. ( _V X. { y } ) ) i^i ran ( 2nd |` ( _V X. _V ) ) ) C_ { ( G ` y ) } -> ( ( { ( G ` y ) } X. ( _V X. { y } ) ) o. ( 2nd |` ( _V X. _V ) ) ) = U_ x e. { ( G ` y ) } ( ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) X. ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) ) )
12 10 11 ax-mp 
 |-  ( ( { ( G ` y ) } X. ( _V X. { y } ) ) o. ( 2nd |` ( _V X. _V ) ) ) = U_ x e. { ( G ` y ) } ( ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) X. ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) )
13 fvex 
 |-  ( G ` y ) e. _V
14 fparlem2 
 |-  ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) = ( _V X. { x } )
15 sneq 
 |-  ( x = ( G ` y ) -> { x } = { ( G ` y ) } )
16 15 xpeq2d 
 |-  ( x = ( G ` y ) -> ( _V X. { x } ) = ( _V X. { ( G ` y ) } ) )
17 14 16 eqtrid 
 |-  ( x = ( G ` y ) -> ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) = ( _V X. { ( G ` y ) } ) )
18 15 imaeq2d 
 |-  ( x = ( G ` y ) -> ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) = ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { ( G ` y ) } ) )
19 df-ima 
 |-  ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { ( G ` y ) } ) = ran ( ( { ( G ` y ) } X. ( _V X. { y } ) ) |` { ( G ` y ) } )
20 ssid 
 |-  { ( G ` y ) } C_ { ( G ` y ) }
21 xpssres 
 |-  ( { ( G ` y ) } C_ { ( G ` y ) } -> ( ( { ( G ` y ) } X. ( _V X. { y } ) ) |` { ( G ` y ) } ) = ( { ( G ` y ) } X. ( _V X. { y } ) ) )
22 20 21 ax-mp 
 |-  ( ( { ( G ` y ) } X. ( _V X. { y } ) ) |` { ( G ` y ) } ) = ( { ( G ` y ) } X. ( _V X. { y } ) )
23 22 rneqi 
 |-  ran ( ( { ( G ` y ) } X. ( _V X. { y } ) ) |` { ( G ` y ) } ) = ran ( { ( G ` y ) } X. ( _V X. { y } ) )
24 13 snnz 
 |-  { ( G ` y ) } =/= (/)
25 rnxp 
 |-  ( { ( G ` y ) } =/= (/) -> ran ( { ( G ` y ) } X. ( _V X. { y } ) ) = ( _V X. { y } ) )
26 24 25 ax-mp 
 |-  ran ( { ( G ` y ) } X. ( _V X. { y } ) ) = ( _V X. { y } )
27 23 26 eqtri 
 |-  ran ( ( { ( G ` y ) } X. ( _V X. { y } ) ) |` { ( G ` y ) } ) = ( _V X. { y } )
28 19 27 eqtri 
 |-  ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { ( G ` y ) } ) = ( _V X. { y } )
29 18 28 eqtrdi 
 |-  ( x = ( G ` y ) -> ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) = ( _V X. { y } ) )
30 17 29 xpeq12d 
 |-  ( x = ( G ` y ) -> ( ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) X. ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) ) = ( ( _V X. { ( G ` y ) } ) X. ( _V X. { y } ) ) )
31 13 30 iunxsn 
 |-  U_ x e. { ( G ` y ) } ( ( `' ( 2nd |` ( _V X. _V ) ) " { x } ) X. ( ( { ( G ` y ) } X. ( _V X. { y } ) ) " { x } ) ) = ( ( _V X. { ( G ` y ) } ) X. ( _V X. { y } ) )
32 12 31 eqtri 
 |-  ( ( { ( G ` y ) } X. ( _V X. { y } ) ) o. ( 2nd |` ( _V X. _V ) ) ) = ( ( _V X. { ( G ` y ) } ) X. ( _V X. { y } ) )
33 32 cnveqi 
 |-  `' ( ( { ( G ` y ) } X. ( _V X. { y } ) ) o. ( 2nd |` ( _V X. _V ) ) ) = `' ( ( _V X. { ( G ` y ) } ) X. ( _V X. { y } ) )
34 cnvco 
 |-  `' ( ( { ( G ` y ) } X. ( _V X. { y } ) ) o. ( 2nd |` ( _V X. _V ) ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. `' ( { ( G ` y ) } X. ( _V X. { y } ) ) )
35 cnvxp 
 |-  `' ( ( _V X. { ( G ` y ) } ) X. ( _V X. { y } ) ) = ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) )
36 33 34 35 3eqtr3i 
 |-  ( `' ( 2nd |` ( _V X. _V ) ) o. `' ( { ( G ` y ) } X. ( _V X. { y } ) ) ) = ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) )
37 fparlem2 
 |-  ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) = ( _V X. { y } )
38 37 xpeq2i 
 |-  ( { ( G ` y ) } X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) = ( { ( G ` y ) } X. ( _V X. { y } ) )
39 fnsnfv 
 |-  ( ( G Fn B /\ y e. B ) -> { ( G ` y ) } = ( G " { y } ) )
40 39 xpeq1d 
 |-  ( ( G Fn B /\ y e. B ) -> ( { ( G ` y ) } X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) = ( ( G " { y } ) X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) )
41 38 40 eqtr3id 
 |-  ( ( G Fn B /\ y e. B ) -> ( { ( G ` y ) } X. ( _V X. { y } ) ) = ( ( G " { y } ) X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) )
42 41 cnveqd 
 |-  ( ( G Fn B /\ y e. B ) -> `' ( { ( G ` y ) } X. ( _V X. { y } ) ) = `' ( ( G " { y } ) X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) )
43 cnvxp 
 |-  `' ( ( G " { y } ) X. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) ) = ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) )
44 42 43 eqtrdi 
 |-  ( ( G Fn B /\ y e. B ) -> `' ( { ( G ` y ) } X. ( _V X. { y } ) ) = ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) )
45 44 coeq2d 
 |-  ( ( G Fn B /\ y e. B ) -> ( `' ( 2nd |` ( _V X. _V ) ) o. `' ( { ( G ` y ) } X. ( _V X. { y } ) ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) ) )
46 36 45 eqtr3id 
 |-  ( ( G Fn B /\ y e. B ) -> ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) ) = ( `' ( 2nd |` ( _V X. _V ) ) o. ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) ) )
47 46 iuneq2dv 
 |-  ( G Fn B -> U_ y e. B ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) ) = U_ y e. B ( `' ( 2nd |` ( _V X. _V ) ) o. ( ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) X. ( G " { y } ) ) ) )
48 1 7 47 3eqtr4a 
 |-  ( G Fn B -> ( `' ( 2nd |` ( _V X. _V ) ) o. ( G o. ( 2nd |` ( _V X. _V ) ) ) ) = U_ y e. B ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) ) )