Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1stpreima Structured version   Visualization version   Unicode version
Theorem 1stpreima 29484
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 681 . . . . . . 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 3597 . . . . . . . 8 |- ( A C_ B -> ( ( 1st ` w ) e. A -> ( 1st ` w ) e. B ) )
43pm4.71d 666 . . . . . . 7 |- ( A C_ B -> ( ( 1st ` w ) e. A <-> ( ( 1st ` w ) e. A /\ ( 1st ` w ) e. B ) ) )
54anbi1d 741 . . . . . 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 838 . . . . . . . 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 730 . . . . . . 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 300 . . . . 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 7201 . . . . . 6 |- ( w e. ( B X. C ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. B /\ ( 2nd ` w ) e. C ) ) )
1110anbi2i 730 . . . . 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 279 . . . 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 838 . . . 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 276 . . 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 5623 . . . . 5 |- ( `' ( 1st |` ( B X. C ) ) " A ) = ( ( `' 1st " A ) i^i ( B X. C ) )
1615eleq2i 2693 . . . 4 |- ( w e. ( `' ( 1st |` ( B X. C ) ) " A ) <-> w e. ( ( `' 1st " A ) i^i ( B X. C ) ) )
17 elin 3796 . . . 4 |- ( w e. ( ( `' 1st " A ) i^i ( B X. C ) ) <-> ( w e. ( `' 1st " A ) /\ w e. ( B X. C ) ) )
18 vex 3203 . . . . . 6 |- w e. _V
19 fo1st 7188 . . . . . . 7 |- 1st : _V -onto-> _V
20 fofn 6117 . . . . . . 7 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
21 elpreima 6337 . . . . . . 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 953 . . . . 5 |- ( w e. ( `' 1st " A ) <-> ( 1st ` w ) e. A )
2423anbi1i 731 . . . 4 |- ( ( w e. ( `' 1st " A ) /\ w e. ( B X. C ) ) <-> ( ( 1st ` w ) e. A /\ w e. ( B X. C ) ) )
2516, 17, 243bitri 286 . . 3 |- ( w e. ( `' ( 1st |` ( B X. C ) ) " A ) <-> ( ( 1st ` w ) e. A /\ w e. ( B X. C ) ) )
26 elxp7 7201 . . 3 |- ( w e. ( A X. C ) <-> ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) e. A /\ ( 2nd ` w ) e. C ) ) )
2714, 25, 263bitr4g 303 . 2 |- ( A C_ B -> ( w e. ( `' ( 1st |` ( B X. C ) ) " A ) <-> w e. ( A X. C ) ) )
2827eqrdv 2620 1 |- ( A C_ B -> ( `' ( 1st |` ( B X. C ) ) " A ) = ( A X. C ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888   1stc1st 7166   2ndc2nd 7167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169
This theorem is referenced by:  sxbrsigalem2  30348
  Copyright terms: Public domain W3C validator