Theorem jm2.27dlem1 43458

Description: Lemma for rmydioph 43463. 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 6834	. 2 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = ((𝑏 ↾ (1...𝐵))‘𝐴))
2		jm2.27dlem1.1	. . 3 ⊢ 𝐴 ∈ (1...𝐵)
3		fvres 6854	. . 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 2788	1 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ↾ cres 5627  ‘cfv 6493  (class class class)co 7361  1c1 11033  ...cfz 13455

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5631  df-res 5637  df-iota 6449  df-fv 6501

This theorem is referenced by:  rmydioph  43463  rmxdioph  43465  expdiophlem2  43471