Theorem 2ndvalg 6228

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 4227	. . 3 |- ( A e. _V -> { A } e. _V )
2		rnexg 4942	. . 3 |- ( { A } e. _V -> ran { A } e. _V )
3		uniexg 4485	. . 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 3643	. . . . 5 |- ( x = A -> { x } = { A } )
6	5	rneqd 4906	. . . 4 |- ( x = A -> ran { x } = ran { A } )
7	6	unieqd 3860	. . 3 |- ( x = A -> U. ran { x } = U. ran { A } )
8		df-2nd 6226	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
9	7, 8	fvmptg 5654	. 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 1372   e. wcel 2175   _Vcvv 2771   {csn 3632   U.cuni 3849   ran crn 4675   ` cfv 5270   2ndc2nd 6224

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fv 5278  df-2nd 6226

This theorem is referenced by:  2nd0  6230  op2nd  6232  elxp6  6254