Theorem 2ndvalg 6377

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 4321	. . 3 |- ( A e. _V -> { A } e. _V )
2		rnexg 5047	. . 3 |- ( { A } e. _V -> ran { A } e. _V )
3		uniexg 4585	. . 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 3720	. . . . 5 |- ( x = A -> { x } = { A } )
6	5	rneqd 5011	. . . 4 |- ( x = A -> ran { x } = ran { A } )
7	6	unieqd 3946	. . 3 |- ( x = A -> U. ran { x } = U. ran { A } )
8		df-2nd 6375	. . 3 |- 2nd = ( x e. _V |-> U. ran { x } )
9	7, 8	fvmptg 5781	. 2 |- ( ( A e. _V /\ U. ran { A } e. _V ) -> ( 2nd ` A ) = U. ran { A } )
10	4, 9	mpdan 425	1 |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3709   U.cuni 3935   ran crn 4775   ` cfv 5377   2ndc2nd 6373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fv 5385  df-2nd 6375

This theorem is used by:  2nd0  6379  op2nd  6381  elxp6  6403