Theorem f2ndres 6139

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 2733	. . . . . . . 8 |- y e. _V
2		vex 2733	. . . . . . . 8 |- z e. _V
3	1, 2	op2nda 5095	. . . . . . 7 |- U. ran { <. y , z >. } = z
4	3	eleq1i 2236	. . . . . 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 2556	. . 3 |- A. y e. A A. z e. B U. ran { <. y , z >. } e. B
8		sneq 3594	. . . . . . 7 |- ( x = <. y , z >. -> { x } = { <. y , z >. } )
9	8	rneqd 4840	. . . . . 6 |- ( x = <. y , z >. -> ran { x } = ran { <. y , z >. } )
10	9	unieqd 3807	. . . . 5 |- ( x = <. y , z >. -> U. ran { x } = U. ran { <. y , z >. } )
11	10	eleq1d 2239	. . . 4 |- ( x = <. y , z >. -> ( U. ran { x } e. B <-> U. ran { <. y , z >. } e. B ) )
12	11	ralxp 4754	. . 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 6120	. . . . 5 |- 2nd = ( x e. _V |-> U. ran { x } )
15	14	reseq1i 4887	. . . 4 |- ( 2nd |` ( A X. B ) ) = ( ( x e. _V |-> U. ran { x } ) |` ( A X. B ) )
16		ssv 3169	. . . . 5 |- ( A X. B ) C_ _V
17		resmpt 4939	. . . . 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 2191	. . 3 |- ( 2nd |` ( A X. B ) ) = ( x e. ( A X. B ) |-> U. ran { x } )
20	19	fmpt 5646	. 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 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   C_ wss 3121   {csn 3583   <.cop 3586   U.cuni 3796   |-> cmpt 4050   X. cxp 4609   ran crn 4612   |` cres 4613   -->wf 5194   2ndc2nd 6118

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-2nd 6120

This theorem is referenced by:  fo2ndresm  6141  2ndcof  6143  f2ndf  6205  eucalgcvga  12012  tx2cn  13064