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Theorem mreriincl 14842
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 4392 . . . 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 14838 . . . 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 2548 . 2  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =  (/) )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C
)
6 mress 14837 . . . . . . 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 2867 . . . . 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 4393 . . . 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 14840 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  C )
1612, 13, 14, 15syl3anc 1223 . . 3  |-  ( ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  /\  I  =/=  (/) )  ->  |^|_ y  e.  I  S  e.  C )
1711, 16eqeltrd 2548 . 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 2778 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 1374    e. wcel 1762    =/= wne 2655   A.wral 2807    i^i cin 3468    C_ wss 3469   (/)c0 3778   |^|_ciin 4319   ` cfv 5579  Moorecmre 14826
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-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
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-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-mre 14830
This theorem is referenced by:  acsfn1  14905  acsfn1c  14906  acsfn2  14907  acsfn1p  30742
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