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Theorem fin1a2lem3 10393
Description: Lemma for fin1a2 10406. (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 7412 . 2 (𝑎 = 𝐴 → (2o ·o 𝑎) = (2o ·o 𝐴))
2 fin1a2lem.b . . 3 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
3 oveq2 7412 . . . 4 (𝑥 = 𝑎 → (2o ·o 𝑥) = (2o ·o 𝑎))
43cbvmptv 5260 . . 3 (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
52, 4eqtri 2761 . 2 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
6 ovex 7437 . 2 (2o ·o 𝐴) ∈ V
71, 5, 6fvmpt 6994 1 (𝐴 ∈ ω → (𝐸𝐴) = (2o ·o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cmpt 5230  cfv 6540  (class class class)co 7404  ωcom 7850  2oc2o 8455   ·o comu 8459
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 5298  ax-nul 5305  ax-pr 5426
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407
This theorem is referenced by:  fin1a2lem4  10394  fin1a2lem5  10395  fin1a2lem6  10396
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