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Theorem hstrlem2 23610
Description: Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
hstrlem2.1  |-  S  =  ( x  e.  CH  |->  ( ( proj  h `  x ) `  u
) )
Assertion
Ref Expression
hstrlem2  |-  ( C  e.  CH  ->  ( S `  C )  =  ( ( proj 
h `  C ) `  u ) )
Distinct variable groups:    x, C    x, u
Allowed substitution hints:    C( u)    S( x, u)

Proof of Theorem hstrlem2
StepHypRef Expression
1 fveq2 5668 . . 3  |-  ( x  =  C  ->  ( proj  h `  x )  =  ( proj  h `  C ) )
21fveq1d 5670 . 2  |-  ( x  =  C  ->  (
( proj  h `  x
) `  u )  =  ( ( proj 
h `  C ) `  u ) )
3 hstrlem2.1 . 2  |-  S  =  ( x  e.  CH  |->  ( ( proj  h `  x ) `  u
) )
4 fvex 5682 . 2  |-  ( (
proj  h `  C ) `
 u )  e. 
_V
52, 3, 4fvmpt 5745 1  |-  ( C  e.  CH  ->  ( S `  C )  =  ( ( proj 
h `  C ) `  u ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    e. cmpt 4207   ` cfv 5394   CHcch 22280   proj  hcpjh 22288
This theorem is referenced by:  hstrlem3a  23611  hstrlem4  23613  hstrlem5  23614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402
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