Theorem bafval 30675

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 6840	. . . . 5 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
2	1	rneqd 5893	. . . 4 ⊢ (𝑢 = 𝑈 → ran ( +𝑣 ‘𝑢) = ran ( +𝑣 ‘𝑈))
3		df-ba 30667	. . . 4 ⊢ BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣 ‘𝑢))
4		fvex 6853	. . . . 5 ⊢ ( +𝑣 ‘𝑈) ∈ V
5	4	rnex 7861	. . . 4 ⊢ ran ( +𝑣 ‘𝑈) ∈ V
6	2, 3, 5	fvmpt 6947	. . 3 ⊢ (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈))
7		rn0 5881	. . . . 5 ⊢ ran ∅ = ∅
8	7	eqcomi 2745	. . . 4 ⊢ ∅ = ran ∅
9		fvprc 6832	. . . 4 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ∅)
10		fvprc 6832	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅)
11	10	rneqd 5893	. . . 4 ⊢ (¬ 𝑈 ∈ V → ran ( +𝑣 ‘𝑈) = ran ∅)
12	8, 9, 11	3eqtr4a 2797	. . 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 5892	. 2 ⊢ ran 𝐺 = ran ( +𝑣 ‘𝑈)
17	13, 14, 16	3eqtr4i 2769	1 ⊢ 𝑋 = ran 𝐺

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ∅c0 4273  ran crn 5632  ‘cfv 6498   +_𝑣 cpv 30656  BaseSetcba 30657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fv 6506  df-ba 30667

This theorem is referenced by:  nvi  30685  nvgf  30689  nvsf  30690  nvgcl  30691  nvcom  30692  nvass  30693  nvadd32  30694  nvrcan  30695  nvadd4  30696  nvscl  30697  nvsid  30698  nvsass  30699  nvdi  30701  nvdir  30702  nv2  30703  nvzcl  30705  nv0rid  30706  nv0lid  30707  nv0  30708  nvsz  30709  nvinv  30710  nvinvfval  30711  nvmval  30713  nvmfval  30715  nvnnncan1  30718  nvnegneg  30720  nvrinv  30722  nvlinv  30723  nvaddsub  30726  cnnvba  30750  sspba  30798  isph  30893  phpar  30895  ip0i  30896  ipdirilem  30900  hhba  31238  hhssabloilem  31332  hhshsslem1  31338