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Theorem 2ndpreima 26173
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 3461 . . . . . . . 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 6722 . . . . . 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 5438 . . . . 5  |-  ( `' ( 2nd  |`  ( B  X.  C ) )
" A )  =  ( ( `' 2nd " A )  i^i  ( B  X.  C ) )
1817eleq2i 2532 . . . 4  |-  ( w  e.  ( `' ( 2nd  |`  ( B  X.  C ) ) " A )  <->  w  e.  ( ( `' 2nd " A )  i^i  ( B  X.  C ) ) )
19 elin 3650 . . . 4  |-  ( w  e.  ( ( `' 2nd " A )  i^i  ( B  X.  C ) )  <->  ( w  e.  ( `' 2nd " A
)  /\  w  e.  ( B  X.  C
) ) )
20 vex 3081 . . . . . 6  |-  w  e. 
_V
21 fo2nd 6710 . . . . . . 7  |-  2nd : _V -onto-> _V
22 fofn 5733 . . . . . . 7  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
23 elpreima 5935 . . . . . . 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 909 . . . . 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 6722 . . 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 2451 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 1370    e. wcel 1758   _Vcvv 3078    i^i cin 3438    C_ wss 3439    X. cxp 4949   `'ccnv 4950    |` cres 4953   "cima 4954    Fn wfn 5524   -onto->wfo 5527   ` cfv 5529   1stc1st 6688   2ndc2nd 6689
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 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-1st 6690  df-2nd 6691
This theorem is referenced by:  sxbrsigalem2  26865
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