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Theorem jm2.27dlem1 36594
 Description: Lemma for rmydioph 36599. Subsitution 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)) → (𝑎𝐴) = (𝑏𝐴))
Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏

Proof of Theorem jm2.27dlem1
StepHypRef Expression
1 fveq1 6102 . 2 (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎𝐴) = ((𝑏 ↾ (1...𝐵))‘𝐴))
2 jm2.27dlem1.1 . . 3 𝐴 ∈ (1...𝐵)
3 fvres 6117 . . 3 (𝐴 ∈ (1...𝐵) → ((𝑏 ↾ (1...𝐵))‘𝐴) = (𝑏𝐴))
42, 3ax-mp 5 . 2 ((𝑏 ↾ (1...𝐵))‘𝐴) = (𝑏𝐴)
51, 4syl6eq 2660 1 (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎𝐴) = (𝑏𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  1c1 9816  ...cfz 12197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-res 5050  df-iota 5768  df-fv 5812 This theorem is referenced by:  rmydioph  36599  rmxdioph  36601  expdiophlem2  36607
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