Theorem bafval 29857

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 6892	. . . . 5 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
2	1	rneqd 5938	. . . 4 ⊢ (𝑢 = 𝑈 → ran ( +𝑣 ‘𝑢) = ran ( +𝑣 ‘𝑈))
3		df-ba 29849	. . . 4 ⊢ BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣 ‘𝑢))
4		fvex 6905	. . . . 5 ⊢ ( +𝑣 ‘𝑈) ∈ V
5	4	rnex 7903	. . . 4 ⊢ ran ( +𝑣 ‘𝑈) ∈ V
6	2, 3, 5	fvmpt 6999	. . 3 ⊢ (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈))
7		rn0 5926	. . . . 5 ⊢ ran ∅ = ∅
8	7	eqcomi 2742	. . . 4 ⊢ ∅ = ran ∅
9		fvprc 6884	. . . 4 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ∅)
10		fvprc 6884	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅)
11	10	rneqd 5938	. . . 4 ⊢ (¬ 𝑈 ∈ V → ran ( +𝑣 ‘𝑈) = ran ∅)
12	8, 9, 11	3eqtr4a 2799	. . 3 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈))
13	6, 12	pm2.61i 182	. 2 ⊢ (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈)
14		bafval.1	. 2 ⊢ 𝑋 = (BaseSet‘𝑈)
15		bafval.2	. . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
16	15	rneqi 5937	. 2 ⊢ ran 𝐺 = ran ( +𝑣 ‘𝑈)
17	13, 14, 16	3eqtr4i 2771	1 ⊢ 𝑋 = ran 𝐺

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ∅c0 4323  ran crn 5678  ‘cfv 6544   +_𝑣 cpv 29838  BaseSetcba 29839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-ba 29849

This theorem is referenced by:  nvi  29867  nvgf  29871  nvsf  29872  nvgcl  29873  nvcom  29874  nvass  29875  nvadd32  29876  nvrcan  29877  nvadd4  29878  nvscl  29879  nvsid  29880  nvsass  29881  nvdi  29883  nvdir  29884  nv2  29885  nvzcl  29887  nv0rid  29888  nv0lid  29889  nv0  29890  nvsz  29891  nvinv  29892  nvinvfval  29893  nvmval  29895  nvmfval  29897  nvnnncan1  29900  nvnegneg  29902  nvrinv  29904  nvlinv  29905  nvaddsub  29908  cnnvba  29932  sspba  29980  isph  30075  phpar  30077  ip0i  30078  ipdirilem  30082  hhba  30420  hhssabloilem  30514  hhshsslem1  30520