MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin1a2lem3 Structured version   Visualization version   GIF version

Theorem fin1a2lem3 10158
Description: Lemma for fin1a2 10171. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
fin1a2lem.b 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
Assertion
Ref Expression
fin1a2lem3 (𝐴 ∈ ω → (𝐸𝐴) = (2o ·o 𝐴))

Proof of Theorem fin1a2lem3
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7283 . 2 (𝑎 = 𝐴 → (2o ·o 𝑎) = (2o ·o 𝐴))
2 fin1a2lem.b . . 3 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
3 oveq2 7283 . . . 4 (𝑥 = 𝑎 → (2o ·o 𝑥) = (2o ·o 𝑎))
43cbvmptv 5187 . . 3 (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
52, 4eqtri 2766 . 2 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
6 ovex 7308 . 2 (2o ·o 𝐴) ∈ V
71, 5, 6fvmpt 6875 1 (𝐴 ∈ ω → (𝐸𝐴) = (2o ·o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  cmpt 5157  cfv 6433  (class class class)co 7275  ωcom 7712  2oc2o 8291   ·o comu 8295
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278
This theorem is referenced by:  fin1a2lem4  10159  fin1a2lem5  10160  fin1a2lem6  10161
  Copyright terms: Public domain W3C validator