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Theorem jm2.27dlem1 30546
Description: Lemma for rmydioph 30551. 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 5858 . 2  |-  ( a  =  ( b  |`  ( 1 ... B
) )  ->  (
a `  A )  =  ( ( b  |`  ( 1 ... B
) ) `  A
) )
2 jm2.27dlem1.1 . . 3  |-  A  e.  ( 1 ... B
)
3 fvres 5873 . . 3  |-  ( A  e.  ( 1 ... B )  ->  (
( b  |`  (
1 ... B ) ) `
 A )  =  ( b `  A
) )
42, 3ax-mp 5 . 2  |-  ( ( b  |`  ( 1 ... B ) ) `
 A )  =  ( b `  A
)
51, 4syl6eq 2519 1  |-  ( a  =  ( b  |`  ( 1 ... B
) )  ->  (
a `  A )  =  ( b `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    |` cres 4996   ` cfv 5581  (class class class)co 6277   1c1 9484   ...cfz 11663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-res 5006  df-iota 5544  df-fv 5589
This theorem is referenced by:  rmydioph  30551  rmxdioph  30553  expdiophlem2  30559
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