Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  jm2.27dlem1 Structured version   Unicode version

Theorem jm2.27dlem1 35313
Description: Lemma for rmydioph 35318. 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  e.  ( 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
StepHypRef Expression
1 fveq1 5848 . 2  |-  ( a  =  ( b  |`  ( 1 ... B
) )  ->  (
a `  A )  =  ( ( b  |`  ( 1 ... B
) ) `  A
) )
2 jm2.27dlem1.1 . . 3  |-  A  e.  ( 1 ... B
)
3 fvres 5863 . . 3  |-  ( A  e.  ( 1 ... B )  ->  (
( b  |`  (
1 ... B ) ) `
 A )  =  ( b `  A
) )
42, 3ax-mp 5 . 2  |-  ( ( b  |`  ( 1 ... B ) ) `
 A )  =  ( b `  A
)
51, 4syl6eq 2459 1  |-  ( a  =  ( b  |`  ( 1 ... B
) )  ->  (
a `  A )  =  ( b `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    |` cres 4825   ` cfv 5569  (class class class)co 6278   1c1 9523   ...cfz 11726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-res 4835  df-iota 5533  df-fv 5577
This theorem is referenced by:  rmydioph  35318  rmxdioph  35320  expdiophlem2  35326
  Copyright terms: Public domain W3C validator