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Theorem mreriincl 14656
Description: The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
mreriincl  |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
Distinct variable groups:    y, I    y, X    y, C
Allowed substitution hint:    S( y)

Proof of Theorem mreriincl
StepHypRef Expression
1 riin0 4353 . . . 4  |-  ( I  =  (/)  ->  ( X  i^i  |^|_ y  e.  I  S )  =  X )
21adantl 466 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  =  X )
3 mre1cl 14652 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
43ad2antrr 725 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  X  e.  C )
52, 4eqeltrd 2542 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
6 mress 14651 . . . . . . 7  |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  S  C_  X )
76ex 434 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( S  e.  C  ->  S  C_  X ) )
87ralimdv 2834 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  ( A. y  e.  I  S  e.  C  ->  A. y  e.  I  S  C_  X
) )
98imp 429 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  A. y  e.  I  S  C_  X
)
10 riinn0 4354 . . . 4  |-  ( ( A. y  e.  I  S  C_  X  /\  I  =/=  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  =  |^|_ y  e.  I  S
)
119, 10sylan 471 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  =  |^|_ y  e.  I  S
)
12 simpll 753 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  C  e.  (Moore `  X )
)
13 simpr 461 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  I  =/=  (/) )
14 simplr 754 . . . 4  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  A. y  e.  I  S  e.  C )
15 mreiincl 14654 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  C )
1612, 13, 14, 15syl3anc 1219 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  |^|_ y  e.  I  S  e.  C )
1711, 16eqeltrd 2542 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
185, 17pm2.61dane 2770 1  |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799    i^i cin 3436    C_ wss 3437   (/)c0 3746   |^|_ciin 4281   ` cfv 5527  Moorecmre 14640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-int 4238  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-mre 14644
This theorem is referenced by:  acsfn1  14719  acsfn1c  14720  acsfn2  14721  acsfn1p  29705
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