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Theorem 1stpreima 28290
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 653 . . . . . . 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 3458 . . . . . . . 8  |-  ( A 
C_  B  ->  (
( 1st `  w
)  e.  A  -> 
( 1st `  w
)  e.  B ) )
43pm4.71d 638 . . . . . . 7  |-  ( A 
C_  B  ->  (
( 1st `  w
)  e.  A  <->  ( ( 1st `  w )  e.  A  /\  ( 1st `  w )  e.  B
) ) )
54anbi1d 709 . . . . . 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 804 . . . . . . . 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 698 . . . . . . 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 288 . . . . 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 6841 . . . . . 6  |-  ( w  e.  ( B  X.  C )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  B  /\  ( 2nd `  w
)  e.  C ) ) )
1110anbi2i 698 . . . . 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 267 . . . 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 804 . . . 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 264 . . 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 5343 . . . . 5  |-  ( `' ( 1st  |`  ( B  X.  C ) )
" A )  =  ( ( `' 1st " A )  i^i  ( B  X.  C ) )
1615eleq2i 2499 . . . 4  |-  ( w  e.  ( `' ( 1st  |`  ( B  X.  C ) ) " A )  <->  w  e.  ( ( `' 1st " A )  i^i  ( B  X.  C ) ) )
17 elin 3649 . . . 4  |-  ( w  e.  ( ( `' 1st " A )  i^i  ( B  X.  C ) )  <->  ( w  e.  ( `' 1st " A
)  /\  w  e.  ( B  X.  C
) ) )
18 vex 3083 . . . . . 6  |-  w  e. 
_V
19 fo1st 6828 . . . . . . 7  |-  1st : _V -onto-> _V
20 fofn 5812 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
21 elpreima 6018 . . . . . . 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 926 . . . . 5  |-  ( w  e.  ( `' 1st " A )  <->  ( 1st `  w )  e.  A
)
2423anbi1i 699 . . . 4  |-  ( ( w  e.  ( `' 1st " A )  /\  w  e.  ( B  X.  C ) )  <->  ( ( 1st `  w )  e.  A  /\  w  e.  ( B  X.  C ) ) )
2516, 17, 243bitri 274 . . 3  |-  ( w  e.  ( `' ( 1st  |`  ( B  X.  C ) ) " A )  <->  ( ( 1st `  w )  e.  A  /\  w  e.  ( B  X.  C
) ) )
26 elxp7 6841 . . 3  |-  ( w  e.  ( A  X.  C )  <->  ( w  e.  ( _V  X.  _V )  /\  ( ( 1st `  w )  e.  A  /\  ( 2nd `  w
)  e.  C ) ) )
2714, 25, 263bitr4g 291 . 2  |-  ( A 
C_  B  ->  (
w  e.  ( `' ( 1st  |`  ( B  X.  C ) )
" A )  <->  w  e.  ( A  X.  C
) ) )
2827eqrdv 2419 1  |-  ( A 
C_  B  ->  ( `' ( 1st  |`  ( B  X.  C ) )
" A )  =  ( A  X.  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3080    i^i cin 3435    C_ wss 3436    X. cxp 4851   `'ccnv 4852    |` cres 4855   "cima 4856    Fn wfn 5596   -onto->wfo 5599   ` cfv 5601   1stc1st 6806   2ndc2nd 6807
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 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
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 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-1st 6808  df-2nd 6809
This theorem is referenced by:  sxbrsigalem2  29117
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