Theorem bafval 27459

Description: Value of the function for the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bafval.1	|- X = ( BaseSet ` U )
bafval.2	|- G = ( +v ` U )

Assertion

Ref	Expression
bafval	|- X = ran G

Proof of Theorem bafval

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( u = U -> ( +v ` u ) = ( +v ` U ) )
2	1	rneqd 5353	. . . 4 |- ( u = U -> ran ( +v ` u ) = ran ( +v ` U ) )
3		df-ba 27451	. . . 4 |- BaseSet = ( u e. _V |-> ran ( +v ` u ) )
4		fvex 6201	. . . . 5 |- ( +v ` U ) e. _V
5	4	rnex 7100	. . . 4 |- ran ( +v ` U ) e. _V
6	2, 3, 5	fvmpt 6282	. . 3 |- ( U e. _V -> ( BaseSet ` U ) = ran ( +v ` U ) )
7		rn0 5377	. . . . 5 |- ran (/) = (/)
8	7	eqcomi 2631	. . . 4 |- (/) = ran (/)
9		fvprc 6185	. . . 4 |- ( -. U e. _V -> ( BaseSet ` U ) = (/) )
10		fvprc 6185	. . . . 5 |- ( -. U e. _V -> ( +v ` U ) = (/) )
11	10	rneqd 5353	. . . 4 |- ( -. U e. _V -> ran ( +v ` U ) = ran (/) )
12	8, 9, 11	3eqtr4a 2682	. . 3 |- ( -. U e. _V -> ( BaseSet ` U ) = ran ( +v ` U ) )
13	6, 12	pm2.61i 176	. 2 |- ( BaseSet ` U ) = ran ( +v ` U )
14		bafval.1	. 2 |- X = ( BaseSet ` U )
15		bafval.2	. . 3 |- G = ( +v ` U )
16	15	rneqi 5352	. 2 |- ran G = ran ( +v ` U )
17	13, 14, 16	3eqtr4i 2654	1 |- X = ran G

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ran crn 5115   ` cfv 5888   +vcpv 27440   BaseSetcba 27441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ba 27451

This theorem is referenced by:  nvi  27469  nvgf  27473  nvsf  27474  nvgcl  27475  nvcom  27476  nvass  27477  nvadd32  27478  nvrcan  27479  nvadd4  27480  nvscl  27481  nvsid  27482  nvsass  27483  nvdi  27485  nvdir  27486  nv2  27487  nvzcl  27489  nv0rid  27490  nv0lid  27491  nv0  27492  nvsz  27493  nvinv  27494  nvinvfval  27495  nvmval  27497  nvmfval  27499  nvnnncan1  27502  nvnegneg  27504  nvrinv  27506  nvlinv  27507  nvaddsub  27510  cnnvba  27534  sspba  27582  isph  27677  phpar  27679  ip0i  27680  ipdirilem  27684  hhba  28024  hhssabloilem  28118  hhshsslem1  28124