Theorem fin1a2lem3 10484

Description: Lemma for fin1a2 10497. (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 7429	. 2 ⊢ (𝑎 = 𝐴 → (2o ·o 𝑎) = (2o ·o 𝐴))
2		fin1a2lem.b	. . 3 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
3		oveq2 7429	. . . 4 ⊢ (𝑥 = 𝑎 → (2o ·o 𝑥) = (2o ·o 𝑎))
4	3	cbvmptv 5208	. . 3 ⊢ (𝑥 ∈ ω ↦ (2o ·o 𝑥)) = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
5	2, 4	eqtri 2783	. 2 ⊢ 𝐸 = (𝑎 ∈ ω ↦ (2o ·o 𝑎))
6		ovex 7454	. 2 ⊢ (2o ·o 𝐴) ∈ V
7	1, 5, 6	fvmpt 6994	1 ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ↦ cmpt 5185  ‘cfv 6539  (class class class)co 7421  ωcom 7878  2_oc2o 8474   ·_o comu 8478

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424

This theorem is used by:  fin1a2lem4  10485  fin1a2lem5  10486  fin1a2lem6  10487