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Theorem fin1a2lem3 10442
Description: Lemma for fin1a2 10455. (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 7439 . 2 (𝑎 = 𝐴 → (2o ·o 𝑎) = (2o ·o 𝐴))
2 fin1a2lem.b . . 3 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
3 oveq2 7439 . . . 4 (𝑥 = 𝑎 → (2o ·o 𝑥) = (2o ·o 𝑎))
43cbvmptv 5255 . . 3 (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
52, 4eqtri 2765 . 2 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
6 ovex 7464 . 2 (2o ·o 𝐴) ∈ V
71, 5, 6fvmpt 7016 1 (𝐴 ∈ ω → (𝐸𝐴) = (2o ·o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cmpt 5225  cfv 6561  (class class class)co 7431  ωcom 7887  2oc2o 8500   ·o comu 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434
This theorem is referenced by:  fin1a2lem4  10443  fin1a2lem5  10444  fin1a2lem6  10445
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