Theorem f2ndres 6058

Description: Mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
f2ndres	|- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B

Proof of Theorem f2ndres

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2689	. . . . . . . 8 |- y e. _V
2		vex 2689	. . . . . . . 8 |- z e. _V
3	1, 2	op2nda 5023	. . . . . . 7 |- U. ran { <. y , z >. } = z
4	3	eleq1i 2205	. . . . . 6 |- ( U. ran { <. y , z >. } e. B <-> z e. B )
5	4	biimpri 132	. . . . 5 |- ( z e. B -> U. ran { <. y , z >. } e. B )
6	5	adantl 275	. . . 4 |- ( ( y e. A /\ z e. B ) -> U. ran { <. y , z >. } e. B )
7	6	rgen2 2518	. . 3 |- A. y e. A A. z e. B U. ran { <. y , z >. } e. B
8		sneq 3538	. . . . . . 7 |- ( x = <. y , z >. -> { x } = { <. y , z >. } )
9	8	rneqd 4768	. . . . . 6 |- ( x = <. y , z >. -> ran { x } = ran { <. y , z >. } )
10	9	unieqd 3747	. . . . 5 |- ( x = <. y , z >. -> U. ran { x } = U. ran { <. y , z >. } )
11	10	eleq1d 2208	. . . 4 |- ( x = <. y , z >. -> ( U. ran { x } e. B <-> U. ran { <. y , z >. } e. B ) )
12	11	ralxp 4682	. . 3 |- ( A. x e. ( A X. B ) U. ran { x } e. B <-> A. y e. A A. z e. B U. ran { <. y , z >. } e. B )
13	7, 12	mpbir 145	. 2 |- A. x e. ( A X. B ) U. ran { x } e. B
14		df-2nd 6039	. . . . 5 |- 2nd = ( x e. _V |-> U. ran { x } )
15	14	reseq1i 4815	. . . 4 |- ( 2nd |` ( A X. B ) ) = ( ( x e. _V |-> U. ran { x } ) |` ( A X. B ) )
16		ssv 3119	. . . . 5 |- ( A X. B ) C_ _V
17		resmpt 4867	. . . . 5 |- ( ( A X. B ) C_ _V -> ( ( x e. _V |-> U. ran { x } ) |` ( A X. B ) ) = ( x e. ( A X. B ) |-> U. ran { x } ) )
18	16, 17	ax-mp 5	. . . 4 |- ( ( x e. _V |-> U. ran { x } ) |` ( A X. B ) ) = ( x e. ( A X. B ) |-> U. ran { x } )
19	15, 18	eqtri 2160	. . 3 |- ( 2nd |` ( A X. B ) ) = ( x e. ( A X. B ) |-> U. ran { x } )
20	19	fmpt 5570	. 2 |- ( A. x e. ( A X. B ) U. ran { x } e. B <-> ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B )
21	13, 20	mpbi 144	1 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B

Syntax hints:   = wceq 1331   e. wcel 1480   A.wral 2416   _Vcvv 2686   C_ wss 3071   {csn 3527   <.cop 3530   U.cuni 3736   |-> cmpt 3989   X. cxp 4537   ran crn 4540   |` cres 4541   -->wf 5119   2ndc2nd 6037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-2nd 6039

This theorem is referenced by:  fo2ndresm  6060  2ndcof  6062  f2ndf  6123  eucalgcvga  11739  tx2cn  12439