Theorem jm2.27dlem1 37576

Description: Lemma for rmydioph 37581. 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 6190	. 2 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = ((𝑏 ↾ (1...𝐵))‘𝐴))
2		jm2.27dlem1.1	. . 3 ⊢ 𝐴 ∈ (1...𝐵)
3		fvres 6207	. . 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	syl6eq 2672	1 ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ↾ cres 5116  ‘cfv 5888  (class class class)co 6650  1c1 9937  ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896

This theorem is referenced by:  rmydioph  37581  rmxdioph  37583  expdiophlem2  37589