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Theorem jm2.27dlem1 27102
Description: Lemma for rmydioph 27107. 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 5524 . 2  |-  ( a  =  ( b  |`  ( 1 ... B
) )  ->  (
a `  A )  =  ( ( b  |`  ( 1 ... B
) ) `  A
) )
2 jm2.27dlem1.1 . . 3  |-  A  e.  ( 1 ... B
)
3 fvres 5542 . . 3  |-  ( A  e.  ( 1 ... B )  ->  (
( b  |`  (
1 ... B ) ) `
 A )  =  ( b `  A
) )
42, 3ax-mp 8 . 2  |-  ( ( b  |`  ( 1 ... B ) ) `
 A )  =  ( b `  A
)
51, 4syl6eq 2331 1  |-  ( a  =  ( b  |`  ( 1 ... B
) )  ->  (
a `  A )  =  ( b `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    |` cres 4691   ` cfv 5255  (class class class)co 5858   1c1 8738   ...cfz 10782
This theorem is referenced by:  rmydioph  27107  rmxdioph  27109  expdiophlem2  27115
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-res 4701  df-iota 5219  df-fv 5263
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