Theorem 2ndvalg 5790

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 3956	. . 3 |- ( A e. _V -> { A } e. _V )
2		rnexg 4615	. . 3 |- ( { A } e. _V -> ran { A } e. _V )
3		uniexg 4193	. . 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 3409	. . . . 5 |- ( x = A -> { x } = { A } )
6	5	rneqd 4581	. . . 4 |- ( x = A -> ran { x } = ran { A } )
7	6	unieqd 3612	. . 3 |- ( x = A -> U. ran { x } = U. ran { A } )
8		df-2nd 5788	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
9	7, 8	fvmptg 5269	. 2 |- ( ( A e. _V /\ U. ran { A } e. _V ) -> ( 2nd ` A ) = U. ran { A } )
10	4, 9	mpdan 412	1 |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   _Vcvv 2601   {csn 3398   U.cuni 3601   ran crn 4364   ` cfv 4922   2ndc2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fv 4930  df-2nd 5788

This theorem is referenced by:  2nd0  5792  op2nd  5794  elxp6  5816