Theorem fparlem2 7278

Description: Lemma for fpar 7281. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fparlem2	|- ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) = ( _V X. { y } )

Proof of Theorem fparlem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvres 6207	. . . . . 6 |- ( x e. ( _V X. _V ) -> ( ( 2nd |` ( _V X. _V ) ) ` x ) = ( 2nd ` x ) )
2	1	eqeq1d 2624	. . . . 5 |- ( x e. ( _V X. _V ) -> ( ( ( 2nd |` ( _V X. _V ) ) ` x ) = y <-> ( 2nd ` x ) = y ) )
3		vex 3203	. . . . . . 7 |- y e. _V
4	3	elsn2 4211	. . . . . 6 |- ( ( 2nd ` x ) e. { y } <-> ( 2nd ` x ) = y )
5		fvex 6201	. . . . . . 7 |- ( 1st ` x ) e. _V
6	5	biantrur 527	. . . . . 6 |- ( ( 2nd ` x ) e. { y } <-> ( ( 1st ` x ) e. _V /\ ( 2nd ` x ) e. { y } ) )
7	4, 6	bitr3i 266	. . . . 5 |- ( ( 2nd ` x ) = y <-> ( ( 1st ` x ) e. _V /\ ( 2nd ` x ) e. { y } ) )
8	2, 7	syl6bb 276	. . . 4 |- ( x e. ( _V X. _V ) -> ( ( ( 2nd |` ( _V X. _V ) ) ` x ) = y <-> ( ( 1st ` x ) e. _V /\ ( 2nd ` x ) e. { y } ) ) )
9	8	pm5.32i 669	. . 3 |- ( ( x e. ( _V X. _V ) /\ ( ( 2nd |` ( _V X. _V ) ) ` x ) = y ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. _V /\ ( 2nd ` x ) e. { y } ) ) )
10		f2ndres 7191	. . . 4 |- ( 2nd |` ( _V X. _V ) ) : ( _V X. _V ) --> _V
11		ffn 6045	. . . 4 |- ( ( 2nd |` ( _V X. _V ) ) : ( _V X. _V ) --> _V -> ( 2nd |` ( _V X. _V ) ) Fn ( _V X. _V ) )
12		fniniseg 6338	. . . 4 |- ( ( 2nd |` ( _V X. _V ) ) Fn ( _V X. _V ) -> ( x e. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) <-> ( x e. ( _V X. _V ) /\ ( ( 2nd |` ( _V X. _V ) ) ` x ) = y ) ) )
13	10, 11, 12	mp2b 10	. . 3 |- ( x e. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) <-> ( x e. ( _V X. _V ) /\ ( ( 2nd |` ( _V X. _V ) ) ` x ) = y ) )
14		elxp7 7201	. . 3 |- ( x e. ( _V X. { y } ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. _V /\ ( 2nd ` x ) e. { y } ) ) )
15	9, 13, 14	3bitr4i 292	. 2 |- ( x e. ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) <-> x e. ( _V X. { y } ) )
16	15	eqriv 2619	1 |- ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) = ( _V X. { y } )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  fparlem4  7280