Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1stpreima Structured version   Unicode version

Theorem 1stpreima 27196
Description: The preimage by  1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
Assertion
Ref Expression
1stpreima  |-  ( A 
C_  B  ->  ( `' ( 1st  |`  ( B  X.  C ) )
" A )  =  ( A  X.  C
) )

Proof of Theorem 1stpreima
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 anass 649 . . . . . . 7  |-  ( ( ( ( 1st `  w
)  e.  A  /\  ( 1st `  w )  e.  B )  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w )  e.  C ) )  <-> 
( ( 1st `  w
)  e.  A  /\  ( ( 1st `  w
)  e.  B  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w )  e.  C ) ) ) )
21a1i 11 . . . . . 6  |-  ( A 
C_  B  ->  (
( ( ( 1st `  w )  e.  A  /\  ( 1st `  w
)  e.  B )  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w
)  e.  C ) )  <->  ( ( 1st `  w )  e.  A  /\  ( ( 1st `  w
)  e.  B  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w )  e.  C ) ) ) ) )
3 ssel 3498 . . . . . . . 8  |-  ( A 
C_  B  ->  (
( 1st `  w
)  e.  A  -> 
( 1st `  w
)  e.  B ) )
43pm4.71d 634 . . . . . . 7  |-  ( A 
C_  B  ->  (
( 1st `  w
)  e.  A  <->  ( ( 1st `  w )  e.  A  /\  ( 1st `  w )  e.  B
) ) )
54anbi1d 704 . . . . . 6  |-  ( A 
C_  B  ->  (
( ( 1st `  w
)  e.  A  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w )  e.  C ) )  <-> 
( ( ( 1st `  w )  e.  A  /\  ( 1st `  w
)  e.  B )  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w
)  e.  C ) ) ) )
6 an12 795 . . . . . . . 8  |-  ( ( w  e.  ( _V 
X.  _V )  /\  (
( 1st `  w
)  e.  B  /\  ( 2nd `  w )  e.  C ) )  <-> 
( ( 1st `  w
)  e.  B  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w )  e.  C ) ) )
76anbi2i 694 . . . . . . 7  |-  ( ( ( 1st `  w
)  e.  A  /\  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w
)  e.  B  /\  ( 2nd `  w )  e.  C ) ) )  <->  ( ( 1st `  w )  e.  A  /\  ( ( 1st `  w
)  e.  B  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w )  e.  C ) ) ) )
87a1i 11 . . . . . 6  |-  ( A 
C_  B  ->  (
( ( 1st `  w
)  e.  A  /\  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w
)  e.  B  /\  ( 2nd `  w )  e.  C ) ) )  <->  ( ( 1st `  w )  e.  A  /\  ( ( 1st `  w
)  e.  B  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w )  e.  C ) ) ) ) )
92, 5, 83bitr4d 285 . . . . 5  |-  ( A 
C_  B  ->  (
( ( 1st `  w
)  e.  A  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w )  e.  C ) )  <-> 
( ( 1st `  w
)  e.  A  /\  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w
)  e.  B  /\  ( 2nd `  w )  e.  C ) ) ) ) )
10 elxp7 6814 . . . . . 6  |-  ( w  e.  ( B  X.  C )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  B  /\  ( 2nd `  w
)  e.  C ) ) )
1110anbi2i 694 . . . . 5  |-  ( ( ( 1st `  w
)  e.  A  /\  w  e.  ( B  X.  C ) )  <->  ( ( 1st `  w )  e.  A  /\  ( w  e.  ( _V  X.  _V )  /\  (
( 1st `  w
)  e.  B  /\  ( 2nd `  w )  e.  C ) ) ) )
129, 11syl6rbbr 264 . . . 4  |-  ( A 
C_  B  ->  (
( ( 1st `  w
)  e.  A  /\  w  e.  ( B  X.  C ) )  <->  ( ( 1st `  w )  e.  A  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w )  e.  C ) ) ) )
13 an12 795 . . . 4  |-  ( ( ( 1st `  w
)  e.  A  /\  ( w  e.  ( _V  X.  _V )  /\  ( 2nd `  w )  e.  C ) )  <-> 
( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w
)  e.  A  /\  ( 2nd `  w )  e.  C ) ) )
1412, 13syl6bb 261 . . 3  |-  ( A 
C_  B  ->  (
( ( 1st `  w
)  e.  A  /\  w  e.  ( B  X.  C ) )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  A  /\  ( 2nd `  w
)  e.  C ) ) ) )
15 cnvresima 5494 . . . . 5  |-  ( `' ( 1st  |`  ( B  X.  C ) )
" A )  =  ( ( `' 1st " A )  i^i  ( B  X.  C ) )
1615eleq2i 2545 . . . 4  |-  ( w  e.  ( `' ( 1st  |`  ( B  X.  C ) ) " A )  <->  w  e.  ( ( `' 1st " A )  i^i  ( B  X.  C ) ) )
17 elin 3687 . . . 4  |-  ( w  e.  ( ( `' 1st " A )  i^i  ( B  X.  C ) )  <->  ( w  e.  ( `' 1st " A
)  /\  w  e.  ( B  X.  C
) ) )
18 vex 3116 . . . . . 6  |-  w  e. 
_V
19 fo1st 6801 . . . . . . 7  |-  1st : _V -onto-> _V
20 fofn 5795 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
21 elpreima 5999 . . . . . . 7  |-  ( 1st 
Fn  _V  ->  ( w  e.  ( `' 1st " A )  <->  ( w  e.  _V  /\  ( 1st `  w )  e.  A
) ) )
2219, 20, 21mp2b 10 . . . . . 6  |-  ( w  e.  ( `' 1st " A )  <->  ( w  e.  _V  /\  ( 1st `  w )  e.  A
) )
2318, 22mpbiran 916 . . . . 5  |-  ( w  e.  ( `' 1st " A )  <->  ( 1st `  w )  e.  A
)
2423anbi1i 695 . . . 4  |-  ( ( w  e.  ( `' 1st " A )  /\  w  e.  ( B  X.  C ) )  <->  ( ( 1st `  w )  e.  A  /\  w  e.  ( B  X.  C ) ) )
2516, 17, 243bitri 271 . . 3  |-  ( w  e.  ( `' ( 1st  |`  ( B  X.  C ) ) " A )  <->  ( ( 1st `  w )  e.  A  /\  w  e.  ( B  X.  C
) ) )
26 elxp7 6814 . . 3  |-  ( w  e.  ( A  X.  C )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  A  /\  ( 2nd `  w
)  e.  C ) ) )
2714, 25, 263bitr4g 288 . 2  |-  ( A 
C_  B  ->  (
w  e.  ( `' ( 1st  |`  ( B  X.  C ) )
" A )  <->  w  e.  ( A  X.  C
) ) )
2827eqrdv 2464 1  |-  ( A 
C_  B  ->  ( `' ( 1st  |`  ( B  X.  C ) )
" A )  =  ( A  X.  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476    X. cxp 4997   `'ccnv 4998    |` cres 5001   "cima 5002    Fn wfn 5581   -onto->wfo 5584   ` cfv 5586   1stc1st 6779   2ndc2nd 6780
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-1st 6781  df-2nd 6782
This theorem is referenced by:  sxbrsigalem2  27897
  Copyright terms: Public domain W3C validator