Theorem f2ndres 6332

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 2806	. . . . . . . 8 |- y e. _V
2		vex 2806	. . . . . . . 8 |- z e. _V
3	1, 2	op2nda 5228	. . . . . . 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 2619	. . 3 |- A. y e. A A. z e. B U. ran { <. y , z >. } e. B
8		sneq 3684	. . . . . . 7 |- ( x = <. y , z >. -> { x } = { <. y , z >. } )
9	8	rneqd 4967	. . . . . 6 |- ( x = <. y , z >. -> ran { x } = ran { <. y , z >. } )
10	9	unieqd 3909	. . . . 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 4879	. . 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 6313	. . . . 5 |- 2nd = ( x e. _V |-> U. ran { x } )
15	14	reseq1i 5015	. . . 4 |- ( 2nd |` ( A X. B ) ) = ( ( x e. _V |-> U. ran { x } ) |` ( A X. B ) )
16		ssv 3250	. . . . 5 |- ( A X. B ) C_ _V
17		resmpt 5067	. . . . 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 5805	. 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 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   C_ wss 3201   {csn 3673   <.cop 3676   U.cuni 3898   |-> cmpt 4155   X. cxp 4729   ran crn 4732   |` cres 4733   -->wf 5329   2ndc2nd 6311

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-2nd 6313

This theorem is referenced by:  fo2ndresm  6334  2ndcof  6336  f2ndf  6400  eucalgcvga  12693  tx2cn  15064