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Theorem jm2.27dlem1 27082
Description: Lemma for rmydioph 27087. 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 5729 . 2  |-  ( a  =  ( b  |`  ( 1 ... B
) )  ->  (
a `  A )  =  ( ( b  |`  ( 1 ... B
) ) `  A
) )
2 jm2.27dlem1.1 . . 3  |-  A  e.  ( 1 ... B
)
3 fvres 5747 . . 3  |-  ( A  e.  ( 1 ... B )  ->  (
( b  |`  (
1 ... B ) ) `
 A )  =  ( b `  A
) )
42, 3ax-mp 8 . 2  |-  ( ( b  |`  ( 1 ... B ) ) `
 A )  =  ( b `  A
)
51, 4syl6eq 2486 1  |-  ( a  =  ( b  |`  ( 1 ... B
) )  ->  (
a `  A )  =  ( b `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    |` cres 4882   ` cfv 5456  (class class class)co 6083   1c1 8993   ...cfz 11045
This theorem is referenced by:  rmydioph  27087  rmxdioph  27089  expdiophlem2  27095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-res 4892  df-iota 5420  df-fv 5464
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