Theorem jm2.27dlem1 40831

Description: Lemma for rmydioph 40836. Substitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Hypothesis

Ref	Expression
jm2.27dlem1.1	⊢ 𝐴 ∈ (1...𝐵)

Assertion

Ref	Expression
jm2.27dlem1	⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴))

Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏

Proof of Theorem jm2.27dlem1

Step	Hyp	Ref	Expression
1		fveq1 6773	. 2 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = ((𝑏 ↾ (1...𝐵))‘𝐴))
2		jm2.27dlem1.1	. . 3 ⊢ 𝐴 ∈ (1...𝐵)
3		fvres 6793	. . 3 ⊢ (𝐴 ∈ (1...𝐵) → ((𝑏 ↾ (1...𝐵))‘𝐴) = (𝑏‘𝐴))
4	2, 3	ax-mp 5	. 2 ⊢ ((𝑏 ↾ (1...𝐵))‘𝐴) = (𝑏‘𝐴)
5	1, 4	eqtrdi 2794	1 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   ↾ cres 5591  ‘cfv 6433  (class class class)co 7275  1c1 10872  ...cfz 13239

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-res 5601  df-iota 6391  df-fv 6441

This theorem is referenced by:  rmydioph  40836  rmxdioph  40838  expdiophlem2  40844