Theorem bafval 30934

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 6883	. . . . 5 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
2	1	rneqd 5930	. . . 4 ⊢ (𝑢 = 𝑈 → ran ( +𝑣 ‘𝑢) = ran ( +𝑣 ‘𝑈))
3		df-ba 30926	. . . 4 ⊢ BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣 ‘𝑢))
4		fvex 6896	. . . . 5 ⊢ ( +𝑣 ‘𝑈) ∈ V
5	4	rnex 7908	. . . 4 ⊢ ran ( +𝑣 ‘𝑈) ∈ V
6	2, 3, 5	fvmpt 6991	. . 3 ⊢ (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈))
7		rn0 5918	. . . . 5 ⊢ ran ∅ = ∅
8	7	eqcomi 2772	. . . 4 ⊢ ∅ = ran ∅
9		fvprc 6875	. . . 4 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ∅)
10		fvprc 6875	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅)
11	10	rneqd 5930	. . . 4 ⊢ (¬ 𝑈 ∈ V → ran ( +𝑣 ‘𝑈) = ran ∅)
12	8, 9, 11	3eqtr4a 2824	. . 3 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈))
13	6, 12	pm2.61i 184	. 2 ⊢ (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈)
14		bafval.1	. 2 ⊢ 𝑋 = (BaseSet‘𝑈)
15		bafval.2	. . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
16	15	rneqi 5929	. 2 ⊢ ran 𝐺 = ran ( +𝑣 ‘𝑈)
17	13, 14, 16	3eqtr4i 2796	1 ⊢ 𝑋 = ran 𝐺

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4287  ran crn 5664  ‘cfv 6538   +_𝑣 cpv 30915  BaseSetcba 30916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-ba 30926

This theorem is referenced by:  nvi  30944  nvgf  30948  nvsf  30949  nvgcl  30950  nvcom  30951  nvass  30952  nvadd32  30953  nvrcan  30954  nvadd4  30955  nvscl  30956  nvsid  30957  nvsass  30958  nvdi  30960  nvdir  30961  nv2  30962  nvzcl  30964  nv0rid  30965  nv0lid  30966  nv0  30967  nvsz  30968  nvinv  30969  nvinvfval  30970  nvmval  30972  nvmfval  30974  nvnnncan1  30977  nvnegneg  30979  nvrinv  30981  nvlinv  30982  nvaddsub  30985  cnnvba  31009  sspba  31057  isph  31152  phpar  31154  ip0i  31155  ipdirilem  31159  hhba  31497  hhssabloilem  31591  hhshsslem1  31597