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Theorem 2ndpreima 28234
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 3401 . . . . . . . 8  |-  ( A 
C_  C  ->  (
( 2nd `  w
)  e.  A  -> 
( 2nd `  w
)  e.  C ) )
21pm4.71rd 639 . . . . . . 7  |-  ( A 
C_  C  ->  (
( 2nd `  w
)  e.  A  <->  ( ( 2nd `  w )  e.  C  /\  ( 2nd `  w )  e.  A
) ) )
32anbi2d 708 . . . . . 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 653 . . . . . . . 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 205 . . . . . . 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 653 . . . . . . . 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 699 . . . . . . 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 282 . . . . 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 6784 . . . . . 6  |-  ( w  e.  ( B  X.  C )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  B  /\  ( 2nd `  w
)  e.  C ) ) )
1211anbi1i 699 . . . . 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 267 . . . 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 451 . . . 4  |-  ( ( w  e.  ( B  X.  C )  /\  ( 2nd `  w )  e.  A )  <->  ( ( 2nd `  w )  e.  A  /\  w  e.  ( B  X.  C
) ) )
15 anass 653 . . . 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 290 . . 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 5286 . . . . 5  |-  ( `' ( 2nd  |`  ( B  X.  C ) )
" A )  =  ( ( `' 2nd " A )  i^i  ( B  X.  C ) )
1817eleq2i 2498 . . . 4  |-  ( w  e.  ( `' ( 2nd  |`  ( B  X.  C ) ) " A )  <->  w  e.  ( ( `' 2nd " A )  i^i  ( B  X.  C ) ) )
19 elin 3592 . . . 4  |-  ( w  e.  ( ( `' 2nd " A )  i^i  ( B  X.  C ) )  <->  ( w  e.  ( `' 2nd " A
)  /\  w  e.  ( B  X.  C
) ) )
20 vex 3025 . . . . . 6  |-  w  e. 
_V
21 fo2nd 6772 . . . . . . 7  |-  2nd : _V -onto-> _V
22 fofn 5755 . . . . . . 7  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
23 elpreima 5961 . . . . . . 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 926 . . . . 5  |-  ( w  e.  ( `' 2nd " A )  <->  ( 2nd `  w )  e.  A
)
2625anbi1i 699 . . . 4  |-  ( ( w  e.  ( `' 2nd " A )  /\  w  e.  ( B  X.  C ) )  <->  ( ( 2nd `  w )  e.  A  /\  w  e.  ( B  X.  C ) ) )
2718, 19, 263bitri 274 . . 3  |-  ( w  e.  ( `' ( 2nd  |`  ( B  X.  C ) ) " A )  <->  ( ( 2nd `  w )  e.  A  /\  w  e.  ( B  X.  C
) ) )
28 elxp7 6784 . . 3  |-  ( w  e.  ( B  X.  A )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  B  /\  ( 2nd `  w
)  e.  A ) ) )
2916, 27, 283bitr4g 291 . 2  |-  ( A 
C_  C  ->  (
w  e.  ( `' ( 2nd  |`  ( B  X.  C ) )
" A )  <->  w  e.  ( B  X.  A
) ) )
3029eqrdv 2426 1  |-  ( A 
C_  C  ->  ( `' ( 2nd  |`  ( B  X.  C ) )
" A )  =  ( B  X.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3022    i^i cin 3378    C_ wss 3379    X. cxp 4794   `'ccnv 4795    |` cres 4798   "cima 4799    Fn wfn 5539   -onto->wfo 5542   ` cfv 5544   1stc1st 6749   2ndc2nd 6750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-fo 5550  df-fv 5552  df-1st 6751  df-2nd 6752
This theorem is referenced by:  sxbrsigalem2  29060
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