Theorem 2ndvalg 6081

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 4140	. . 3 |- ( A e. _V -> { A } e. _V )
2		rnexg 4844	. . 3 |- ( { A } e. _V -> ran { A } e. _V )
3		uniexg 4394	. . 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 3567	. . . . 5 |- ( x = A -> { x } = { A } )
6	5	rneqd 4808	. . . 4 |- ( x = A -> ran { x } = ran { A } )
7	6	unieqd 3779	. . 3 |- ( x = A -> U. ran { x } = U. ran { A } )
8		df-2nd 6079	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
9	7, 8	fvmptg 5537	. 2 |- ( ( A e. _V /\ U. ran { A } e. _V ) -> ( 2nd ` A ) = U. ran { A } )
10	4, 9	mpdan 418	1 |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 2125   _Vcvv 2709   {csn 3556   U.cuni 3768   ran crn 4580   ` cfv 5163   2ndc2nd 6077

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-iota 5128  df-fun 5165  df-fv 5171  df-2nd 6079

This theorem is referenced by:  2nd0  6083  op2nd  6085  elxp6  6107