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 → (𝐸‘𝐴) = (2o ·o 𝐴))

Proof of Theorem fin1a2lem3

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	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))
4	3	cbvmptv 5260	. . 3 ⊢ (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
5	2, 4	eqtri 2761	. 2 ⊢ 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
6		ovex 7437	. 2 ⊢ (2o ·o 𝐴) ∈ V
7	1, 5, 6	fvmpt 6994	1 ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5230  ‘cfv 6540  (class class class)co 7404  ωcom 7850  2_oc2o 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