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Theorem fin1a2lem3 9539
 Description: Lemma for fin1a2 9552. (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 6913 . 2 (𝑎 = 𝐴 → (2o ·o 𝑎) = (2o ·o 𝐴))
2 fin1a2lem.b . . 3 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
3 oveq2 6913 . . . 4 (𝑥 = 𝑎 → (2o ·o 𝑥) = (2o ·o 𝑎))
43cbvmptv 4973 . . 3 (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
52, 4eqtri 2849 . 2 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
6 ovex 6937 . 2 (2o ·o 𝐴) ∈ V
71, 5, 6fvmpt 6529 1 (𝐴 ∈ ω → (𝐸𝐴) = (2o ·o 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166   ↦ cmpt 4952  ‘cfv 6123  (class class class)co 6905  ωcom 7326  2oc2o 7820   ·o comu 7824 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908 This theorem is referenced by:  fin1a2lem4  9540  fin1a2lem5  9541  fin1a2lem6  9542
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