Theorem jm2.27dlem1 40358

Description: Lemma for rmydioph 40363. 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 6662	. 2 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = ((𝑏 ↾ (1...𝐵))‘𝐴))
2		jm2.27dlem1.1	. . 3 ⊢ 𝐴 ∈ (1...𝐵)
3		fvres 6682	. . 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 2809	1 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ↾ cres 5530  ‘cfv 6340  (class class class)co 7156  1c1 10589  ...cfz 12952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-xp 5534  df-res 5540  df-iota 6299  df-fv 6348

This theorem is referenced by:  rmydioph  40363  rmxdioph  40365  expdiophlem2  40371