Theorem fin1a2lem3 9810

Description: Lemma for fin1a2 9823. (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 7150	. 2 ⊢ (𝑎 = 𝐴 → (2o ·o 𝑎) = (2o ·o 𝐴))
2		fin1a2lem.b	. . 3 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
3		oveq2 7150	. . . 4 ⊢ (𝑥 = 𝑎 → (2o ·o 𝑥) = (2o ·o 𝑎))
4	3	cbvmptv 5155	. . 3 ⊢ (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
5	2, 4	eqtri 2844	. 2 ⊢ 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
6		ovex 7175	. 2 ⊢ (2o ·o 𝐴) ∈ V
7	1, 5, 6	fvmpt 6754	1 ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ↦ cmpt 5132  ‘cfv 6341  (class class class)co 7142  ωcom 7566  2_oc2o 8082   ·_o comu 8086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pr 5316

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3488  df-sbc 3764  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-iota 6300  df-fun 6343  df-fv 6349  df-ov 7145

This theorem is referenced by:  fin1a2lem4  9811  fin1a2lem5  9812  fin1a2lem6  9813