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Theorem jm2.27dlem1 41738
Description: Lemma for rmydioph 41743. 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 6890 . 2 (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎𝐴) = ((𝑏 ↾ (1...𝐵))‘𝐴))
2 jm2.27dlem1.1 . . 3 𝐴 ∈ (1...𝐵)
3 fvres 6910 . . 3 (𝐴 ∈ (1...𝐵) → ((𝑏 ↾ (1...𝐵))‘𝐴) = (𝑏𝐴))
42, 3ax-mp 5 . 2 ((𝑏 ↾ (1...𝐵))‘𝐴) = (𝑏𝐴)
51, 4eqtrdi 2788 1 (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎𝐴) = (𝑏𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cres 5678  cfv 6543  (class class class)co 7408  1c1 11110  ...cfz 13483
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-res 5688  df-iota 6495  df-fv 6551
This theorem is referenced by:  rmydioph  41743  rmxdioph  41745  expdiophlem2  41751
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