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Theorem isofrlem 6235
Description: Lemma for isofr 6237. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
Hypotheses
Ref Expression
isofrlem.1  |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )
isofrlem.2  |-  ( ph  ->  ( H " x
)  e.  _V )
Assertion
Ref Expression
isofrlem  |-  ( ph  ->  ( S  Fr  B  ->  R  Fr  A ) )
Distinct variable groups:    x, A    x, B    x, H    ph, x    x, R    x, S

Proof of Theorem isofrlem
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isofrlem.1 . . . . . . 7  |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )
2 isof1o 6220 . . . . . . 7  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
31, 2syl 16 . . . . . 6  |-  ( ph  ->  H : A -1-1-onto-> B )
4 f1ofn 5823 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
5 n0 3799 . . . . . . . . . 10  |-  ( x  =/=  (/)  <->  E. y  y  e.  x )
6 fnfvima 6149 . . . . . . . . . . . . 13  |-  ( ( H  Fn  A  /\  x  C_  A  /\  y  e.  x )  ->  ( H `  y )  e.  ( H " x
) )
7 ne0i 3796 . . . . . . . . . . . . 13  |-  ( ( H `  y )  e.  ( H "
x )  ->  ( H " x )  =/=  (/) )
86, 7syl 16 . . . . . . . . . . . 12  |-  ( ( H  Fn  A  /\  x  C_  A  /\  y  e.  x )  ->  ( H " x )  =/=  (/) )
983expia 1198 . . . . . . . . . . 11  |-  ( ( H  Fn  A  /\  x  C_  A )  -> 
( y  e.  x  ->  ( H " x
)  =/=  (/) ) )
109exlimdv 1700 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  x  C_  A )  -> 
( E. y  y  e.  x  ->  ( H " x )  =/=  (/) ) )
115, 10syl5bi 217 . . . . . . . . 9  |-  ( ( H  Fn  A  /\  x  C_  A )  -> 
( x  =/=  (/)  ->  ( H " x )  =/=  (/) ) )
1211expimpd 603 . . . . . . . 8  |-  ( H  Fn  A  ->  (
( x  C_  A  /\  x  =/=  (/) )  -> 
( H " x
)  =/=  (/) ) )
134, 12syl 16 . . . . . . 7  |-  ( H : A -1-1-onto-> B  ->  ( (
x  C_  A  /\  x  =/=  (/) )  ->  ( H " x )  =/=  (/) ) )
14 f1ofo 5829 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H : A -onto-> B )
15 imassrn 5354 . . . . . . . . 9  |-  ( H
" x )  C_  ran  H
16 forn 5804 . . . . . . . . 9  |-  ( H : A -onto-> B  ->  ran  H  =  B )
1715, 16syl5sseq 3557 . . . . . . . 8  |-  ( H : A -onto-> B  -> 
( H " x
)  C_  B )
1814, 17syl 16 . . . . . . 7  |-  ( H : A -1-1-onto-> B  ->  ( H " x )  C_  B
)
1913, 18jctild 543 . . . . . 6  |-  ( H : A -1-1-onto-> B  ->  ( (
x  C_  A  /\  x  =/=  (/) )  ->  (
( H " x
)  C_  B  /\  ( H " x )  =/=  (/) ) ) )
203, 19syl 16 . . . . 5  |-  ( ph  ->  ( ( x  C_  A  /\  x  =/=  (/) )  -> 
( ( H "
x )  C_  B  /\  ( H " x
)  =/=  (/) ) ) )
21 dffr3 5375 . . . . . 6  |-  ( S  Fr  B  <->  A. z
( ( z  C_  B  /\  z  =/=  (/) )  ->  E. w  e.  z 
( z  i^i  ( `' S " { w } ) )  =  (/) ) )
22 isofrlem.2 . . . . . . 7  |-  ( ph  ->  ( H " x
)  e.  _V )
23 sseq1 3530 . . . . . . . . . 10  |-  ( z  =  ( H "
x )  ->  (
z  C_  B  <->  ( H " x )  C_  B
) )
24 neeq1 2748 . . . . . . . . . 10  |-  ( z  =  ( H "
x )  ->  (
z  =/=  (/)  <->  ( H " x )  =/=  (/) ) )
2523, 24anbi12d 710 . . . . . . . . 9  |-  ( z  =  ( H "
x )  ->  (
( z  C_  B  /\  z  =/=  (/) )  <->  ( ( H " x )  C_  B  /\  ( H "
x )  =/=  (/) ) ) )
26 ineq1 3698 . . . . . . . . . . 11  |-  ( z  =  ( H "
x )  ->  (
z  i^i  ( `' S " { w }
) )  =  ( ( H " x
)  i^i  ( `' S " { w }
) ) )
2726eqeq1d 2469 . . . . . . . . . 10  |-  ( z  =  ( H "
x )  ->  (
( z  i^i  ( `' S " { w } ) )  =  (/) 
<->  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )
2827rexeqbi1dv 3072 . . . . . . . . 9  |-  ( z  =  ( H "
x )  ->  ( E. w  e.  z 
( z  i^i  ( `' S " { w } ) )  =  (/) 
<->  E. w  e.  ( H " x ) ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )
2925, 28imbi12d 320 . . . . . . . 8  |-  ( z  =  ( H "
x )  ->  (
( ( z  C_  B  /\  z  =/=  (/) )  ->  E. w  e.  z 
( z  i^i  ( `' S " { w } ) )  =  (/) )  <->  ( ( ( H " x ) 
C_  B  /\  ( H " x )  =/=  (/) )  ->  E. w  e.  ( H " x
) ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/) ) ) )
3029spcgv 3203 . . . . . . 7  |-  ( ( H " x )  e.  _V  ->  ( A. z ( ( z 
C_  B  /\  z  =/=  (/) )  ->  E. w  e.  z  ( z  i^i  ( `' S " { w } ) )  =  (/) )  -> 
( ( ( H
" x )  C_  B  /\  ( H "
x )  =/=  (/) )  ->  E. w  e.  ( H " x ) ( ( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) ) )
3122, 30syl 16 . . . . . 6  |-  ( ph  ->  ( A. z ( ( z  C_  B  /\  z  =/=  (/) )  ->  E. w  e.  z 
( z  i^i  ( `' S " { w } ) )  =  (/) )  ->  ( ( ( H " x
)  C_  B  /\  ( H " x )  =/=  (/) )  ->  E. w  e.  ( H " x
) ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/) ) ) )
3221, 31syl5bi 217 . . . . 5  |-  ( ph  ->  ( S  Fr  B  ->  ( ( ( H
" x )  C_  B  /\  ( H "
x )  =/=  (/) )  ->  E. w  e.  ( H " x ) ( ( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) ) )
3320, 32syl5d 67 . . . 4  |-  ( ph  ->  ( S  Fr  B  ->  ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. w  e.  ( H " x ) ( ( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) ) )
343adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  A
)  ->  H : A
-1-1-onto-> B )
35 f1ofun 5824 . . . . . . . . . . 11  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
3634, 35syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  A
)  ->  Fun  H )
37 simpl 457 . . . . . . . . . 10  |-  ( ( w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) )  ->  w  e.  ( H " x
) )
38 fvelima 5926 . . . . . . . . . 10  |-  ( ( Fun  H  /\  w  e.  ( H " x
) )  ->  E. y  e.  x  ( H `  y )  =  w )
3936, 37, 38syl2an 477 . . . . . . . . 9  |-  ( ( ( ph  /\  x  C_  A )  /\  (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )  ->  E. y  e.  x  ( H `  y )  =  w )
40 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) )  ->  ( ( H " x )  i^i  ( `' S " { w } ) )  =  (/) )
41 ssel 3503 . . . . . . . . . . . . . . . . . . 19  |-  ( x 
C_  A  ->  (
y  e.  x  -> 
y  e.  A ) )
4241imdistani 690 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  C_  A  /\  y  e.  x )  ->  ( x  C_  A  /\  y  e.  A
) )
43 isomin 6232 . . . . . . . . . . . . . . . . . 18  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  C_  A  /\  y  e.  A )
)  ->  ( (
x  i^i  ( `' R " { y } ) )  =  (/)  <->  (
( H " x
)  i^i  ( `' S " { ( H `
 y ) } ) )  =  (/) ) )
441, 42, 43syl2an 477 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( x  C_  A  /\  y  e.  x ) )  -> 
( ( x  i^i  ( `' R " { y } ) )  =  (/)  <->  ( ( H " x )  i^i  ( `' S " { ( H `  y ) } ) )  =  (/) ) )
45 sneq 4043 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( H `  y )  =  w  ->  { ( H `  y ) }  =  { w } )
4645imaeq2d 5343 . . . . . . . . . . . . . . . . . . 19  |-  ( ( H `  y )  =  w  ->  ( `' S " { ( H `  y ) } )  =  ( `' S " { w } ) )
4746ineq2d 3705 . . . . . . . . . . . . . . . . . 18  |-  ( ( H `  y )  =  w  ->  (
( H " x
)  i^i  ( `' S " { ( H `
 y ) } ) )  =  ( ( H " x
)  i^i  ( `' S " { w }
) ) )
4847eqeq1d 2469 . . . . . . . . . . . . . . . . 17  |-  ( ( H `  y )  =  w  ->  (
( ( H "
x )  i^i  ( `' S " { ( H `  y ) } ) )  =  (/) 
<->  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )
4944, 48sylan9bb 699 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  C_  A  /\  y  e.  x )
)  /\  ( H `  y )  =  w )  ->  ( (
x  i^i  ( `' R " { y } ) )  =  (/)  <->  (
( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) )
5040, 49syl5ibr 221 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  A  /\  y  e.  x )
)  /\  ( H `  y )  =  w )  ->  ( (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) )  ->  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
5150exp42 611 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  C_  A  ->  ( y  e.  x  ->  ( ( H `  y )  =  w  ->  ( ( w  e.  ( H "
x )  /\  (
( H " x
)  i^i  ( `' S " { w }
) )  =  (/) )  ->  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) ) ) )
5251imp 429 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  C_  A
)  ->  ( y  e.  x  ->  ( ( H `  y )  =  w  ->  (
( w  e.  ( H " x )  /\  ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/) )  -> 
( x  i^i  ( `' R " { y } ) )  =  (/) ) ) ) )
5352com3l 81 . . . . . . . . . . . 12  |-  ( y  e.  x  ->  (
( H `  y
)  =  w  -> 
( ( ph  /\  x  C_  A )  -> 
( ( w  e.  ( H " x
)  /\  ( ( H " x )  i^i  ( `' S " { w } ) )  =  (/) )  -> 
( x  i^i  ( `' R " { y } ) )  =  (/) ) ) ) )
5453com4t 85 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  A
)  ->  ( (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) )  ->  ( y  e.  x  ->  (
( H `  y
)  =  w  -> 
( x  i^i  ( `' R " { y } ) )  =  (/) ) ) ) )
5554imp 429 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  C_  A )  /\  (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )  ->  (
y  e.  x  -> 
( ( H `  y )  =  w  ->  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
5655reximdvai 2939 . . . . . . . . 9  |-  ( ( ( ph  /\  x  C_  A )  /\  (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )  ->  ( E. y  e.  x  ( H `  y )  =  w  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
5739, 56mpd 15 . . . . . . . 8  |-  ( ( ( ph  /\  x  C_  A )  /\  (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) )
5857rexlimdvaa 2960 . . . . . . 7  |-  ( (
ph  /\  x  C_  A
)  ->  ( E. w  e.  ( H " x ) ( ( H " x )  i^i  ( `' S " { w } ) )  =  (/)  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
5958ex 434 . . . . . 6  |-  ( ph  ->  ( x  C_  A  ->  ( E. w  e.  ( H " x
) ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/)  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
6059adantrd 468 . . . . 5  |-  ( ph  ->  ( ( x  C_  A  /\  x  =/=  (/) )  -> 
( E. w  e.  ( H " x
) ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/)  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
6160a2d 26 . . . 4  |-  ( ph  ->  ( ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. w  e.  ( H " x
) ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/) )  -> 
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
6233, 61syld 44 . . 3  |-  ( ph  ->  ( S  Fr  B  ->  ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
6362alrimdv 1697 . 2  |-  ( ph  ->  ( S  Fr  B  ->  A. x ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
64 dffr3 5375 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
6563, 64syl6ibr 227 1  |-  ( ph  ->  ( S  Fr  B  ->  R  Fr  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   {csn 4033    Fr wfr 4841   `'ccnv 5004   ran crn 5006   "cima 5008   Fun wfun 5588    Fn wfn 5589   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-fr 4844  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603
This theorem is referenced by:  isofr  6237  isofr2  6239  isowe2  6245
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