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Theorem 1stpreima 27673
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 3493 . . . . . . . 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 797 . . . . . . . 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 6832 . . . . . 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 797 . . . 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 5502 . . . . 5  |-  ( `' ( 1st  |`  ( B  X.  C ) )
" A )  =  ( ( `' 1st " A )  i^i  ( B  X.  C ) )
1615eleq2i 2535 . . . 4  |-  ( w  e.  ( `' ( 1st  |`  ( B  X.  C ) ) " A )  <->  w  e.  ( ( `' 1st " A )  i^i  ( B  X.  C ) ) )
17 elin 3683 . . . 4  |-  ( w  e.  ( ( `' 1st " A )  i^i  ( B  X.  C ) )  <->  ( w  e.  ( `' 1st " A
)  /\  w  e.  ( B  X.  C
) ) )
18 vex 3112 . . . . . 6  |-  w  e. 
_V
19 fo1st 6819 . . . . . . 7  |-  1st : _V -onto-> _V
20 fofn 5803 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
21 elpreima 6008 . . . . . . 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 918 . . . . 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 6832 . . 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 2454 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 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    C_ wss 3471    X. cxp 5006   `'ccnv 5007    |` cres 5010   "cima 5011    Fn wfn 5589   -onto->wfo 5592   ` cfv 5594   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-1st 6799  df-2nd 6800
This theorem is referenced by:  sxbrsigalem2  28418
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