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Theorem isofrlem 5853
Description: Lemma for isofr 5855. (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 5838 . . . . . . 7  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
31, 2syl 15 . . . . . 6  |-  ( ph  ->  H : A -1-1-onto-> B )
4 f1ofn 5489 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
5 n0 3477 . . . . . . . . . 10  |-  ( x  =/=  (/)  <->  E. y  y  e.  x )
6 fnfvima 5772 . . . . . . . . . . . . 13  |-  ( ( H  Fn  A  /\  x  C_  A  /\  y  e.  x )  ->  ( H `  y )  e.  ( H " x
) )
7 ne0i 3474 . . . . . . . . . . . . 13  |-  ( ( H `  y )  e.  ( H "
x )  ->  ( H " x )  =/=  (/) )
86, 7syl 15 . . . . . . . . . . . 12  |-  ( ( H  Fn  A  /\  x  C_  A  /\  y  e.  x )  ->  ( H " x )  =/=  (/) )
983expia 1153 . . . . . . . . . . 11  |-  ( ( H  Fn  A  /\  x  C_  A )  -> 
( y  e.  x  ->  ( H " x
)  =/=  (/) ) )
109exlimdv 1626 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  x  C_  A )  -> 
( E. y  y  e.  x  ->  ( H " x )  =/=  (/) ) )
115, 10syl5bi 208 . . . . . . . . 9  |-  ( ( H  Fn  A  /\  x  C_  A )  -> 
( x  =/=  (/)  ->  ( H " x )  =/=  (/) ) )
1211expimpd 586 . . . . . . . 8  |-  ( H  Fn  A  ->  (
( x  C_  A  /\  x  =/=  (/) )  -> 
( H " x
)  =/=  (/) ) )
134, 12syl 15 . . . . . . 7  |-  ( H : A -1-1-onto-> B  ->  ( (
x  C_  A  /\  x  =/=  (/) )  ->  ( H " x )  =/=  (/) ) )
14 f1ofo 5495 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H : A -onto-> B )
15 imassrn 5041 . . . . . . . . 9  |-  ( H
" x )  C_  ran  H
16 forn 5470 . . . . . . . . 9  |-  ( H : A -onto-> B  ->  ran  H  =  B )
1715, 16syl5sseq 3239 . . . . . . . 8  |-  ( H : A -onto-> B  -> 
( H " x
)  C_  B )
1814, 17syl 15 . . . . . . 7  |-  ( H : A -1-1-onto-> B  ->  ( H " x )  C_  B
)
1913, 18jctild 527 . . . . . 6  |-  ( H : A -1-1-onto-> B  ->  ( (
x  C_  A  /\  x  =/=  (/) )  ->  (
( H " x
)  C_  B  /\  ( H " x )  =/=  (/) ) ) )
203, 19syl 15 . . . . 5  |-  ( ph  ->  ( ( x  C_  A  /\  x  =/=  (/) )  -> 
( ( H "
x )  C_  B  /\  ( H " x
)  =/=  (/) ) ) )
21 dffr3 5061 . . . . . 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 3212 . . . . . . . . . 10  |-  ( z  =  ( H "
x )  ->  (
z  C_  B  <->  ( H " x )  C_  B
) )
24 neeq1 2467 . . . . . . . . . 10  |-  ( z  =  ( H "
x )  ->  (
z  =/=  (/)  <->  ( H " x )  =/=  (/) ) )
2523, 24anbi12d 691 . . . . . . . . 9  |-  ( z  =  ( H "
x )  ->  (
( z  C_  B  /\  z  =/=  (/) )  <->  ( ( H " x )  C_  B  /\  ( H "
x )  =/=  (/) ) ) )
26 ineq1 3376 . . . . . . . . . . 11  |-  ( z  =  ( H "
x )  ->  (
z  i^i  ( `' S " { w }
) )  =  ( ( H " x
)  i^i  ( `' S " { w }
) ) )
2726eqeq1d 2304 . . . . . . . . . 10  |-  ( z  =  ( H "
x )  ->  (
( z  i^i  ( `' S " { w } ) )  =  (/) 
<->  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )
2827rexeqbi1dv 2758 . . . . . . . . 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 311 . . . . . . . 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 2881 . . . . . . 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 15 . . . . . 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 208 . . . . 5  |-  ( ph  ->  ( S  Fr  B  ->  ( ( ( H
" x )  C_  B  /\  ( H "
x )  =/=  (/) )  ->  E. w  e.  ( H " x ) ( ( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) ) )
3320, 32syl5d 62 . . . 4  |-  ( ph  ->  ( S  Fr  B  ->  ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. w  e.  ( H " x ) ( ( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) ) )
343adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  A
)  ->  H : A
-1-1-onto-> B )
35 f1ofun 5490 . . . . . . . . . . . 12  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
3634, 35syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  A
)  ->  Fun  H )
37 simpl 443 . . . . . . . . . . 11  |-  ( ( w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) )  ->  w  e.  ( H " x
) )
38 fvelima 5590 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  w  e.  ( H " x
) )  ->  E. y  e.  x  ( H `  y )  =  w )
3936, 37, 38syl2an 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  C_  A )  /\  (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )  ->  E. y  e.  x  ( H `  y )  =  w )
40 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( ( w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) )  ->  ( ( H " x )  i^i  ( `' S " { w } ) )  =  (/) )
41 ssel 3187 . . . . . . . . . . . . . . . . . . . 20  |-  ( x 
C_  A  ->  (
y  e.  x  -> 
y  e.  A ) )
4241imdistani 671 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  C_  A  /\  y  e.  x )  ->  ( x  C_  A  /\  y  e.  A
) )
43 isomin 5850 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 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 463 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( x  C_  A  /\  y  e.  x ) )  -> 
( ( x  i^i  ( `' R " { y } ) )  =  (/)  <->  ( ( H " x )  i^i  ( `' S " { ( H `  y ) } ) )  =  (/) ) )
45 sneq 3664 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( H `  y )  =  w  ->  { ( H `  y ) }  =  { w } )
4645imaeq2d 5028 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( H `  y )  =  w  ->  ( `' S " { ( H `  y ) } )  =  ( `' S " { w } ) )
4746ineq2d 3383 . . . . . . . . . . . . . . . . . . 19  |-  ( ( H `  y )  =  w  ->  (
( H " x
)  i^i  ( `' S " { ( H `
 y ) } ) )  =  ( ( H " x
)  i^i  ( `' S " { w }
) ) )
4847eqeq1d 2304 . . . . . . . . . . . . . . . . . 18  |-  ( ( H `  y )  =  w  ->  (
( ( H "
x )  i^i  ( `' S " { ( H `  y ) } ) )  =  (/) 
<->  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )
4944, 48sylan9bb 680 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  C_  A  /\  y  e.  x )
)  /\  ( H `  y )  =  w )  ->  ( (
x  i^i  ( `' R " { y } ) )  =  (/)  <->  (
( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) )
5040, 49syl5ibr 212 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 594 . . . . . . . . . . . . . . 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 } ) )  =  (/) ) ) ) ) )
5251imp 418 . . . . . . . . . . . . . 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 } ) )  =  (/) ) ) ) )
5352com3l 75 . . . . . . . . . . . . 13  |-  ( 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 79 . . . . . . . . . . . 12  |-  ( (
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 418 . . . . . . . . . . 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 } ) )  =  (/) ) ) )
5655reximdvai 2666 . . . . . . . . . 10  |-  ( ( ( 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 14 . . . . . . . . 9  |-  ( ( ( ph  /\  x  C_  A )  /\  (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) )
5857exp32 588 . . . . . . . 8  |-  ( (
ph  /\  x  C_  A
)  ->  ( w  e.  ( H " x
)  ->  ( (
( H " x
)  i^i  ( `' S " { w }
) )  =  (/)  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
5958rexlimdv 2679 . . . . . . 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 } ) )  =  (/) ) )
6059ex 423 . . . . . 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 } ) )  =  (/) ) ) )
6160adantrd 454 . . . . 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 } ) )  =  (/) ) ) )
6261a2d 23 . . . 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 } ) )  =  (/) ) ) )
6333, 62syld 40 . . 3  |-  ( ph  ->  ( S  Fr  B  ->  ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
6463alrimdv 1623 . 2  |-  ( ph  ->  ( S  Fr  B  ->  A. x ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
65 dffr3 5061 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
6664, 65syl6ibr 218 1  |-  ( ph  ->  ( S  Fr  B  ->  R  Fr  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653    Fr wfr 4365   `'ccnv 4704   ran crn 4706   "cima 4708   Fun wfun 5265    Fn wfn 5266   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272
This theorem is referenced by:  isofr  5855  isofr2  5857  isowe2  5863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-fr 4368  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280
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