Theorem 2ndvalg 6289

Description: The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
2ndvalg	|- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )

Proof of Theorem 2ndvalg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snexg 4268	. . 3 |- ( A e. _V -> { A } e. _V )
2		rnexg 4989	. . 3 |- ( { A } e. _V -> ran { A } e. _V )
3		uniexg 4530	. . 3 |- ( ran { A } e. _V -> U. ran { A } e. _V )
4	1, 2, 3	3syl 17	. 2 |- ( A e. _V -> U. ran { A } e. _V )
5		sneq 3677	. . . . 5 |- ( x = A -> { x } = { A } )
6	5	rneqd 4953	. . . 4 |- ( x = A -> ran { x } = ran { A } )
7	6	unieqd 3899	. . 3 |- ( x = A -> U. ran { x } = U. ran { A } )
8		df-2nd 6287	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
9	7, 8	fvmptg 5710	. 2 |- ( ( A e. _V /\ U. ran { A } e. _V ) -> ( 2nd ` A ) = U. ran { A } )
10	4, 9	mpdan 421	1 |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   U.cuni 3888   ran crn 4720   ` cfv 5318   2ndc2nd 6285

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fv 5326  df-2nd 6287

This theorem is referenced by:  2nd0  6291  op2nd  6293  elxp6  6315