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Theorem 2ndpreima 27348
Description: The preimage by  2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
Assertion
Ref Expression
2ndpreima  |-  ( A 
C_  C  ->  ( `' ( 2nd  |`  ( B  X.  C ) )
" A )  =  ( B  X.  A
) )

Proof of Theorem 2ndpreima
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ssel 3503 . . . . . . . 8  |-  ( A 
C_  C  ->  (
( 2nd `  w
)  e.  A  -> 
( 2nd `  w
)  e.  C ) )
21pm4.71rd 635 . . . . . . 7  |-  ( A 
C_  C  ->  (
( 2nd `  w
)  e.  A  <->  ( ( 2nd `  w )  e.  C  /\  ( 2nd `  w )  e.  A
) ) )
32anbi2d 703 . . . . . 6  |-  ( A 
C_  C  ->  (
( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( 2nd `  w
)  e.  A )  <-> 
( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( ( 2nd `  w )  e.  C  /\  ( 2nd `  w
)  e.  A ) ) ) )
4 anass 649 . . . . . . . 8  |-  ( ( ( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( 2nd `  w
)  e.  C )  /\  ( 2nd `  w
)  e.  A )  <-> 
( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( ( 2nd `  w )  e.  C  /\  ( 2nd `  w
)  e.  A ) ) )
54bicomi 202 . . . . . . 7  |-  ( ( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( ( 2nd `  w )  e.  C  /\  ( 2nd `  w
)  e.  A ) )  <->  ( ( ( w  e.  ( _V 
X.  _V )  /\  ( 1st `  w )  e.  B )  /\  ( 2nd `  w )  e.  C )  /\  ( 2nd `  w )  e.  A ) )
65a1i 11 . . . . . 6  |-  ( A 
C_  C  ->  (
( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( ( 2nd `  w )  e.  C  /\  ( 2nd `  w
)  e.  A ) )  <->  ( ( ( w  e.  ( _V 
X.  _V )  /\  ( 1st `  w )  e.  B )  /\  ( 2nd `  w )  e.  C )  /\  ( 2nd `  w )  e.  A ) ) )
7 anass 649 . . . . . . . 8  |-  ( ( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( 2nd `  w
)  e.  C )  <-> 
( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w
)  e.  B  /\  ( 2nd `  w )  e.  C ) ) )
87anbi1i 695 . . . . . . 7  |-  ( ( ( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( 2nd `  w
)  e.  C )  /\  ( 2nd `  w
)  e.  A )  <-> 
( ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  B  /\  ( 2nd `  w
)  e.  C ) )  /\  ( 2nd `  w )  e.  A
) )
98a1i 11 . . . . . 6  |-  ( A 
C_  C  ->  (
( ( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w )  e.  B )  /\  ( 2nd `  w )  e.  C )  /\  ( 2nd `  w )  e.  A )  <->  ( (
w  e.  ( _V 
X.  _V )  /\  (
( 1st `  w
)  e.  B  /\  ( 2nd `  w )  e.  C ) )  /\  ( 2nd `  w
)  e.  A ) ) )
103, 6, 93bitrd 279 . . . . 5  |-  ( A 
C_  C  ->  (
( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( 2nd `  w
)  e.  A )  <-> 
( ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  B  /\  ( 2nd `  w
)  e.  C ) )  /\  ( 2nd `  w )  e.  A
) ) )
11 elxp7 6828 . . . . . 6  |-  ( w  e.  ( B  X.  C )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  B  /\  ( 2nd `  w
)  e.  C ) ) )
1211anbi1i 695 . . . . 5  |-  ( ( w  e.  ( B  X.  C )  /\  ( 2nd `  w )  e.  A )  <->  ( (
w  e.  ( _V 
X.  _V )  /\  (
( 1st `  w
)  e.  B  /\  ( 2nd `  w )  e.  C ) )  /\  ( 2nd `  w
)  e.  A ) )
1310, 12syl6rbbr 264 . . . 4  |-  ( A 
C_  C  ->  (
( w  e.  ( B  X.  C )  /\  ( 2nd `  w
)  e.  A )  <-> 
( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( 2nd `  w
)  e.  A ) ) )
14 ancom 450 . . . 4  |-  ( ( w  e.  ( B  X.  C )  /\  ( 2nd `  w )  e.  A )  <->  ( ( 2nd `  w )  e.  A  /\  w  e.  ( B  X.  C
) ) )
15 anass 649 . . . 4  |-  ( ( ( w  e.  ( _V  X.  _V )  /\  ( 1st `  w
)  e.  B )  /\  ( 2nd `  w
)  e.  A )  <-> 
( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w
)  e.  B  /\  ( 2nd `  w )  e.  A ) ) )
1613, 14, 153bitr3g 287 . . 3  |-  ( A 
C_  C  ->  (
( ( 2nd `  w
)  e.  A  /\  w  e.  ( B  X.  C ) )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  B  /\  ( 2nd `  w
)  e.  A ) ) ) )
17 cnvresima 5502 . . . . 5  |-  ( `' ( 2nd  |`  ( B  X.  C ) )
" A )  =  ( ( `' 2nd " A )  i^i  ( B  X.  C ) )
1817eleq2i 2545 . . . 4  |-  ( w  e.  ( `' ( 2nd  |`  ( B  X.  C ) ) " A )  <->  w  e.  ( ( `' 2nd " A )  i^i  ( B  X.  C ) ) )
19 elin 3692 . . . 4  |-  ( w  e.  ( ( `' 2nd " A )  i^i  ( B  X.  C ) )  <->  ( w  e.  ( `' 2nd " A
)  /\  w  e.  ( B  X.  C
) ) )
20 vex 3121 . . . . . 6  |-  w  e. 
_V
21 fo2nd 6816 . . . . . . 7  |-  2nd : _V -onto-> _V
22 fofn 5803 . . . . . . 7  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
23 elpreima 6008 . . . . . . 7  |-  ( 2nd 
Fn  _V  ->  ( w  e.  ( `' 2nd " A )  <->  ( w  e.  _V  /\  ( 2nd `  w )  e.  A
) ) )
2421, 22, 23mp2b 10 . . . . . 6  |-  ( w  e.  ( `' 2nd " A )  <->  ( w  e.  _V  /\  ( 2nd `  w )  e.  A
) )
2520, 24mpbiran 916 . . . . 5  |-  ( w  e.  ( `' 2nd " A )  <->  ( 2nd `  w )  e.  A
)
2625anbi1i 695 . . . 4  |-  ( ( w  e.  ( `' 2nd " A )  /\  w  e.  ( B  X.  C ) )  <->  ( ( 2nd `  w )  e.  A  /\  w  e.  ( B  X.  C ) ) )
2718, 19, 263bitri 271 . . 3  |-  ( w  e.  ( `' ( 2nd  |`  ( B  X.  C ) ) " A )  <->  ( ( 2nd `  w )  e.  A  /\  w  e.  ( B  X.  C
) ) )
28 elxp7 6828 . . 3  |-  ( w  e.  ( B  X.  A )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  B  /\  ( 2nd `  w
)  e.  A ) ) )
2916, 27, 283bitr4g 288 . 2  |-  ( A 
C_  C  ->  (
w  e.  ( `' ( 2nd  |`  ( B  X.  C ) )
" A )  <->  w  e.  ( B  X.  A
) ) )
3029eqrdv 2464 1  |-  ( A 
C_  C  ->  ( `' ( 2nd  |`  ( B  X.  C ) )
" A )  =  ( B  X.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480    C_ wss 3481    X. cxp 5003   `'ccnv 5004    |` cres 5007   "cima 5008    Fn wfn 5589   -onto->wfo 5592   ` cfv 5594   1stc1st 6793   2ndc2nd 6794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-1st 6795  df-2nd 6796
This theorem is referenced by:  sxbrsigalem2  28082
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