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Theorem 1stpreima 26145
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 3451 . . . . . . . 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 6712 . . . . . 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 5428 . . . . 5  |-  ( `' ( 1st  |`  ( B  X.  C ) )
" A )  =  ( ( `' 1st " A )  i^i  ( B  X.  C ) )
1615eleq2i 2529 . . . 4  |-  ( w  e.  ( `' ( 1st  |`  ( B  X.  C ) ) " A )  <->  w  e.  ( ( `' 1st " A )  i^i  ( B  X.  C ) ) )
17 elin 3640 . . . 4  |-  ( w  e.  ( ( `' 1st " A )  i^i  ( B  X.  C ) )  <->  ( w  e.  ( `' 1st " A
)  /\  w  e.  ( B  X.  C
) ) )
18 vex 3074 . . . . . 6  |-  w  e. 
_V
19 fo1st 6699 . . . . . . 7  |-  1st : _V -onto-> _V
20 fofn 5723 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
21 elpreima 5925 . . . . . . 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 909 . . . . 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 6712 . . 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 2448 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 1370    e. wcel 1758   _Vcvv 3071    i^i cin 3428    C_ wss 3429    X. cxp 4939   `'ccnv 4940    |` cres 4943   "cima 4944    Fn wfn 5514   -onto->wfo 5517   ` cfv 5519   1stc1st 6678   2ndc2nd 6679
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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fo 5525  df-fv 5527  df-1st 6680  df-2nd 6681
This theorem is referenced by:  sxbrsigalem2  26838
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