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Theorem jm2.27dlem1 41376
Description: Lemma for rmydioph 41381. 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)) → (𝑎𝐴) = (𝑏𝐴))
Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏

Proof of Theorem jm2.27dlem1
StepHypRef Expression
1 fveq1 6842 . 2 (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎𝐴) = ((𝑏 ↾ (1...𝐵))‘𝐴))
2 jm2.27dlem1.1 . . 3 𝐴 ∈ (1...𝐵)
3 fvres 6862 . . 3 (𝐴 ∈ (1...𝐵) → ((𝑏 ↾ (1...𝐵))‘𝐴) = (𝑏𝐴))
42, 3ax-mp 5 . 2 ((𝑏 ↾ (1...𝐵))‘𝐴) = (𝑏𝐴)
51, 4eqtrdi 2789 1 (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎𝐴) = (𝑏𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cres 5636  cfv 6497  (class class class)co 7358  1c1 11057  ...cfz 13430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-res 5646  df-iota 6449  df-fv 6505
This theorem is referenced by:  rmydioph  41381  rmxdioph  41383  expdiophlem2  41389
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