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Theorem fin1a2lem3 9895
Description: Lemma for fin1a2 9908. (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 7172 . 2 (𝑎 = 𝐴 → (2o ·o 𝑎) = (2o ·o 𝐴))
2 fin1a2lem.b . . 3 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
3 oveq2 7172 . . . 4 (𝑥 = 𝑎 → (2o ·o 𝑥) = (2o ·o 𝑎))
43cbvmptv 5130 . . 3 (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
52, 4eqtri 2761 . 2 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
6 ovex 7197 . 2 (2o ·o 𝐴) ∈ V
71, 5, 6fvmpt 6769 1 (𝐴 ∈ ω → (𝐸𝐴) = (2o ·o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2113  cmpt 5107  cfv 6333  (class class class)co 7164  ωcom 7593  2oc2o 8118   ·o comu 8122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fv 6341  df-ov 7167
This theorem is referenced by:  fin1a2lem4  9896  fin1a2lem5  9897  fin1a2lem6  9898
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