Theorem fin1a2lem3 10387

Description: Lemma for fin1a2 10400. (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 7420	. 2 ⊢ (𝑎 = 𝐴 → (2o ·o 𝑎) = (2o ·o 𝐴))
2		fin1a2lem.b	. . 3 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
3		oveq2 7420	. . . 4 ⊢ (𝑥 = 𝑎 → (2o ·o 𝑥) = (2o ·o 𝑎))
4	3	cbvmptv 5216	. . 3 ⊢ (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
5	2, 4	eqtri 2786	. 2 ⊢ 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
6		ovex 7445	. 2 ⊢ (2o ·o 𝐴) ∈ V
7	1, 5, 6	fvmpt 6991	1 ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143   ↦ cmpt 5193  ‘cfv 6538  (class class class)co 7412  ωcom 7863  2_oc2o 8448   ·_o comu 8452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415

This theorem is referenced by:  fin1a2lem4  10388  fin1a2lem5  10389  fin1a2lem6  10390