Theorem f2ndres 6322

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 2805	. . . . . . . 8 |- y e. _V
2		vex 2805	. . . . . . . 8 |- z e. _V
3	1, 2	op2nda 5221	. . . . . . 7 |- U. ran { <. y , z >. } = z
4	3	eleq1i 2297	. . . . . 6 |- ( U. ran { <. y , z >. } e. B <-> z e. B )
5	4	biimpri 133	. . . . 5 |- ( z e. B -> U. ran { <. y , z >. } e. B )
6	5	adantl 277	. . . 4 |- ( ( y e. A /\ z e. B ) -> U. ran { <. y , z >. } e. B )
7	6	rgen2 2618	. . 3 |- A. y e. A A. z e. B U. ran { <. y , z >. } e. B
8		sneq 3680	. . . . . . 7 |- ( x = <. y , z >. -> { x } = { <. y , z >. } )
9	8	rneqd 4961	. . . . . 6 |- ( x = <. y , z >. -> ran { x } = ran { <. y , z >. } )
10	9	unieqd 3904	. . . . 5 |- ( x = <. y , z >. -> U. ran { x } = U. ran { <. y , z >. } )
11	10	eleq1d 2300	. . . 4 |- ( x = <. y , z >. -> ( U. ran { x } e. B <-> U. ran { <. y , z >. } e. B ) )
12	11	ralxp 4873	. . 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 146	. 2 |- A. x e. ( A X. B ) U. ran { x } e. B
14		df-2nd 6303	. . . . 5 |- 2nd = ( x e. _V |-> U. ran { x } )
15	14	reseq1i 5009	. . . 4 |- ( 2nd |` ( A X. B ) ) = ( ( x e. _V |-> U. ran { x } ) |` ( A X. B ) )
16		ssv 3249	. . . . 5 |- ( A X. B ) C_ _V
17		resmpt 5061	. . . . 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 2252	. . 3 |- ( 2nd |` ( A X. B ) ) = ( x e. ( A X. B ) |-> U. ran { x } )
20	19	fmpt 5797	. 2 |- ( A. x e. ( A X. B ) U. ran { x } e. B <-> ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B )
21	13, 20	mpbi 145	1 |- ( 2nd |` ( A X. B ) ) : ( A X. B ) --> B

Syntax hints:   = wceq 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   {csn 3669   <.cop 3672   U.cuni 3893   |-> cmpt 4150   X. cxp 4723   ran crn 4726   |` cres 4727   -->wf 5322   2ndc2nd 6301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-2nd 6303

This theorem is referenced by:  fo2ndresm  6324  2ndcof  6326  f2ndf  6390  eucalgcvga  12629  tx2cn  14993