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Theorem bafval 29595
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 = (BaseSetβ€˜π‘ˆ)
bafval.2 𝐺 = ( +𝑣 β€˜π‘ˆ)
Assertion
Ref Expression
bafval 𝑋 = ran 𝐺

Proof of Theorem bafval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6846 . . . . 5 (𝑒 = π‘ˆ β†’ ( +𝑣 β€˜π‘’) = ( +𝑣 β€˜π‘ˆ))
21rneqd 5897 . . . 4 (𝑒 = π‘ˆ β†’ ran ( +𝑣 β€˜π‘’) = ran ( +𝑣 β€˜π‘ˆ))
3 df-ba 29587 . . . 4 BaseSet = (𝑒 ∈ V ↦ ran ( +𝑣 β€˜π‘’))
4 fvex 6859 . . . . 5 ( +𝑣 β€˜π‘ˆ) ∈ V
54rnex 7853 . . . 4 ran ( +𝑣 β€˜π‘ˆ) ∈ V
62, 3, 5fvmpt 6952 . . 3 (π‘ˆ ∈ V β†’ (BaseSetβ€˜π‘ˆ) = ran ( +𝑣 β€˜π‘ˆ))
7 rn0 5885 . . . . 5 ran βˆ… = βˆ…
87eqcomi 2742 . . . 4 βˆ… = ran βˆ…
9 fvprc 6838 . . . 4 (Β¬ π‘ˆ ∈ V β†’ (BaseSetβ€˜π‘ˆ) = βˆ…)
10 fvprc 6838 . . . . 5 (Β¬ π‘ˆ ∈ V β†’ ( +𝑣 β€˜π‘ˆ) = βˆ…)
1110rneqd 5897 . . . 4 (Β¬ π‘ˆ ∈ V β†’ ran ( +𝑣 β€˜π‘ˆ) = ran βˆ…)
128, 9, 113eqtr4a 2799 . . 3 (Β¬ π‘ˆ ∈ V β†’ (BaseSetβ€˜π‘ˆ) = ran ( +𝑣 β€˜π‘ˆ))
136, 12pm2.61i 182 . 2 (BaseSetβ€˜π‘ˆ) = ran ( +𝑣 β€˜π‘ˆ)
14 bafval.1 . 2 𝑋 = (BaseSetβ€˜π‘ˆ)
15 bafval.2 . . 3 𝐺 = ( +𝑣 β€˜π‘ˆ)
1615rneqi 5896 . 2 ran 𝐺 = ran ( +𝑣 β€˜π‘ˆ)
1713, 14, 163eqtr4i 2771 1 𝑋 = ran 𝐺
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1542   ∈ wcel 2107  Vcvv 3447  βˆ…c0 4286  ran crn 5638  β€˜cfv 6500   +𝑣 cpv 29576  BaseSetcba 29577
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 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fv 6508  df-ba 29587
This theorem is referenced by:  nvi  29605  nvgf  29609  nvsf  29610  nvgcl  29611  nvcom  29612  nvass  29613  nvadd32  29614  nvrcan  29615  nvadd4  29616  nvscl  29617  nvsid  29618  nvsass  29619  nvdi  29621  nvdir  29622  nv2  29623  nvzcl  29625  nv0rid  29626  nv0lid  29627  nv0  29628  nvsz  29629  nvinv  29630  nvinvfval  29631  nvmval  29633  nvmfval  29635  nvnnncan1  29638  nvnegneg  29640  nvrinv  29642  nvlinv  29643  nvaddsub  29646  cnnvba  29670  sspba  29718  isph  29813  phpar  29815  ip0i  29816  ipdirilem  29820  hhba  30158  hhssabloilem  30252  hhshsslem1  30258
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