Theorem f2ndres 6128

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 2729	. . . . . . . 8 |- y e. _V
2		vex 2729	. . . . . . . 8 |- z e. _V
3	1, 2	op2nda 5088	. . . . . . 7 |- U. ran { <. y , z >. } = z
4	3	eleq1i 2232	. . . . . 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 2552	. . 3 |- A. y e. A A. z e. B U. ran { <. y , z >. } e. B
8		sneq 3587	. . . . . . 7 |- ( x = <. y , z >. -> { x } = { <. y , z >. } )
9	8	rneqd 4833	. . . . . 6 |- ( x = <. y , z >. -> ran { x } = ran { <. y , z >. } )
10	9	unieqd 3800	. . . . 5 |- ( x = <. y , z >. -> U. ran { x } = U. ran { <. y , z >. } )
11	10	eleq1d 2235	. . . 4 |- ( x = <. y , z >. -> ( U. ran { x } e. B <-> U. ran { <. y , z >. } e. B ) )
12	11	ralxp 4747	. . 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 6109	. . . . 5 |- 2nd = ( x e. _V |-> U. ran { x } )
15	14	reseq1i 4880	. . . 4 |- ( 2nd |` ( A X. B ) ) = ( ( x e. _V |-> U. ran { x } ) |` ( A X. B ) )
16		ssv 3164	. . . . 5 |- ( A X. B ) C_ _V
17		resmpt 4932	. . . . 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 2186	. . 3 |- ( 2nd |` ( A X. B ) ) = ( x e. ( A X. B ) |-> U. ran { x } )
20	19	fmpt 5635	. 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 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   C_ wss 3116   {csn 3576   <.cop 3579   U.cuni 3789   |-> cmpt 4043   X. cxp 4602   ran crn 4605   |` cres 4606   -->wf 5184   2ndc2nd 6107

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-2nd 6109

This theorem is referenced by:  fo2ndresm  6130  2ndcof  6132  f2ndf  6194  eucalgcvga  11990  tx2cn  12910