Theorem bafval 30585

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 6876	. . . . 5 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
2	1	rneqd 5918	. . . 4 ⊢ (𝑢 = 𝑈 → ran ( +𝑣 ‘𝑢) = ran ( +𝑣 ‘𝑈))
3		df-ba 30577	. . . 4 ⊢ BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣 ‘𝑢))
4		fvex 6889	. . . . 5 ⊢ ( +𝑣 ‘𝑈) ∈ V
5	4	rnex 7906	. . . 4 ⊢ ran ( +𝑣 ‘𝑈) ∈ V
6	2, 3, 5	fvmpt 6986	. . 3 ⊢ (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈))
7		rn0 5905	. . . . 5 ⊢ ran ∅ = ∅
8	7	eqcomi 2744	. . . 4 ⊢ ∅ = ran ∅
9		fvprc 6868	. . . 4 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ∅)
10		fvprc 6868	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅)
11	10	rneqd 5918	. . . 4 ⊢ (¬ 𝑈 ∈ V → ran ( +𝑣 ‘𝑈) = ran ∅)
12	8, 9, 11	3eqtr4a 2796	. . 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 5917	. 2 ⊢ ran 𝐺 = ran ( +𝑣 ‘𝑈)
17	13, 14, 16	3eqtr4i 2768	1 ⊢ 𝑋 = ran 𝐺

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ∅c0 4308  ran crn 5655  ‘cfv 6531   +_𝑣 cpv 30566  BaseSetcba 30567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fv 6539  df-ba 30577

This theorem is referenced by:  nvi  30595  nvgf  30599  nvsf  30600  nvgcl  30601  nvcom  30602  nvass  30603  nvadd32  30604  nvrcan  30605  nvadd4  30606  nvscl  30607  nvsid  30608  nvsass  30609  nvdi  30611  nvdir  30612  nv2  30613  nvzcl  30615  nv0rid  30616  nv0lid  30617  nv0  30618  nvsz  30619  nvinv  30620  nvinvfval  30621  nvmval  30623  nvmfval  30625  nvnnncan1  30628  nvnegneg  30630  nvrinv  30632  nvlinv  30633  nvaddsub  30636  cnnvba  30660  sspba  30708  isph  30803  phpar  30805  ip0i  30806  ipdirilem  30810  hhba  31148  hhssabloilem  31242  hhshsslem1  31248