Theorem jm2.27dlem1 43627

Description: Lemma for rmydioph 43632. 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 6881	. 2 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = ((𝑏 ↾ (1...𝐵))‘𝐴))
2		jm2.27dlem1.1	. . 3 ⊢ 𝐴 ∈ (1...𝐵)
3		fvres 6901	. . 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 2820	1 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149   ↾ cres 5664  ‘cfv 6537  (class class class)co 7411  1c1 11100  ...cfz 13534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-res 5674  df-iota 6493  df-fv 6545

This theorem is referenced by:  rmydioph  43632  rmxdioph  43634  expdiophlem2  43640