Theorem bafval 30753

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 6863	. . . . 5 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
2	1	rneqd 5912	. . . 4 ⊢ (𝑢 = 𝑈 → ran ( +𝑣 ‘𝑢) = ran ( +𝑣 ‘𝑈))
3		df-ba 30745	. . . 4 ⊢ BaseSet = (𝑢 ∈ V ↦ ran ( +𝑣 ‘𝑢))
4		fvex 6876	. . . . 5 ⊢ ( +𝑣 ‘𝑈) ∈ V
5	4	rnex 7887	. . . 4 ⊢ ran ( +𝑣 ‘𝑈) ∈ V
6	2, 3, 5	fvmpt 6971	. . 3 ⊢ (𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈))
7		rn0 5900	. . . . 5 ⊢ ran ∅ = ∅
8	7	eqcomi 2770	. . . 4 ⊢ ∅ = ran ∅
9		fvprc 6855	. . . 4 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ∅)
10		fvprc 6855	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅)
11	10	rneqd 5912	. . . 4 ⊢ (¬ 𝑈 ∈ V → ran ( +𝑣 ‘𝑈) = ran ∅)
12	8, 9, 11	3eqtr4a 2822	. . 3 ⊢ (¬ 𝑈 ∈ V → (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈))
13	6, 12	pm2.61i 183	. 2 ⊢ (BaseSet‘𝑈) = ran ( +𝑣 ‘𝑈)
14		bafval.1	. 2 ⊢ 𝑋 = (BaseSet‘𝑈)
15		bafval.2	. . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
16	15	rneqi 5911	. 2 ⊢ ran 𝐺 = ran ( +𝑣 ‘𝑈)
17	13, 14, 16	3eqtr4i 2794	1 ⊢ 𝑋 = ran 𝐺

Syntax hints:  ¬ wn 3   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ∅c0 4285  ran crn 5646  ‘cfv 6517   +_𝑣 cpv 30734  BaseSetcba 30735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fv 6525  df-ba 30745

This theorem is referenced by:  nvi  30763  nvgf  30767  nvsf  30768  nvgcl  30769  nvcom  30770  nvass  30771  nvadd32  30772  nvrcan  30773  nvadd4  30774  nvscl  30775  nvsid  30776  nvsass  30777  nvdi  30779  nvdir  30780  nv2  30781  nvzcl  30783  nv0rid  30784  nv0lid  30785  nv0  30786  nvsz  30787  nvinv  30788  nvinvfval  30789  nvmval  30791  nvmfval  30793  nvnnncan1  30796  nvnegneg  30798  nvrinv  30800  nvlinv  30801  nvaddsub  30804  cnnvba  30828  sspba  30876  isph  30971  phpar  30973  ip0i  30974  ipdirilem  30978  hhba  31316  hhssabloilem  31410  hhshsslem1  31416