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Theorem isoselem 5838
Description: Lemma for isose 5840. (Contributed by Mario Carneiro, 23-Jun-2015.)
Hypotheses
Ref Expression
isofrlem.1  |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )
isofrlem.2  |-  ( ph  ->  ( H " x
)  e.  _V )
Assertion
Ref Expression
isoselem  |-  ( ph  ->  ( R Se  A  ->  S Se  B ) )
Distinct variable groups:    x, A    x, B    x, H    ph, x    x, R    x, S

Proof of Theorem isoselem
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfse2 5046 . . . . . . . . 9  |-  ( R Se  A  <->  A. z  e.  A  ( A  i^i  ( `' R " { z } ) )  e. 
_V )
21biimpi 186 . . . . . . . 8  |-  ( R Se  A  ->  A. z  e.  A  ( A  i^i  ( `' R " { z } ) )  e.  _V )
32r19.21bi 2641 . . . . . . 7  |-  ( ( R Se  A  /\  z  e.  A )  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
43expcom 424 . . . . . 6  |-  ( z  e.  A  ->  ( R Se  A  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
)
54adantl 452 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  ( R Se  A  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
)
6 imaeq2 5008 . . . . . . . . . . 11  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( H " x
)  =  ( H
" ( A  i^i  ( `' R " { z } ) ) ) )
76eleq1d 2349 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( ( H "
x )  e.  _V  <->  ( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
87imbi2d 307 . . . . . . . . 9  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( ( ph  ->  ( H " x )  e.  _V )  <->  ( ph  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V ) ) )
9 isofrlem.2 . . . . . . . . 9  |-  ( ph  ->  ( H " x
)  e.  _V )
108, 9vtoclg 2843 . . . . . . . 8  |-  ( ( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( ph  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
1110com12 27 . . . . . . 7  |-  ( ph  ->  ( ( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( H "
( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
1211adantr 451 . . . . . 6  |-  ( (
ph  /\  z  e.  A )  ->  (
( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( H "
( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
13 isofrlem.1 . . . . . . . 8  |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )
14 isoini 5835 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  z  e.  A )  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
1513, 14sylan 457 . . . . . . 7  |-  ( (
ph  /\  z  e.  A )  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
1615eleq1d 2349 . . . . . 6  |-  ( (
ph  /\  z  e.  A )  ->  (
( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V  <->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
1712, 16sylibd 205 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  (
( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
185, 17syld 40 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  ( R Se  A  ->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e.  _V )
)
1918ralrimdva 2633 . . 3  |-  ( ph  ->  ( R Se  A  ->  A. z  e.  A  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
20 isof1o 5822 . . . . . 6  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
2113, 20syl 15 . . . . 5  |-  ( ph  ->  H : A -1-1-onto-> B )
22 f1ofn 5473 . . . . 5  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
23 sneq 3651 . . . . . . . . 9  |-  ( y  =  ( H `  z )  ->  { y }  =  { ( H `  z ) } )
2423imaeq2d 5012 . . . . . . . 8  |-  ( y  =  ( H `  z )  ->  ( `' S " { y } )  =  ( `' S " { ( H `  z ) } ) )
2524ineq2d 3370 . . . . . . 7  |-  ( y  =  ( H `  z )  ->  ( B  i^i  ( `' S " { y } ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
2625eleq1d 2349 . . . . . 6  |-  ( y  =  ( H `  z )  ->  (
( B  i^i  ( `' S " { y } ) )  e. 
_V 
<->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
2726ralrn 5668 . . . . 5  |-  ( H  Fn  A  ->  ( A. y  e.  ran  H ( B  i^i  ( `' S " { y } ) )  e. 
_V 
<-> 
A. z  e.  A  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
2821, 22, 273syl 18 . . . 4  |-  ( ph  ->  ( A. y  e. 
ran  H ( B  i^i  ( `' S " { y } ) )  e.  _V  <->  A. z  e.  A  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e.  _V )
)
29 f1ofo 5479 . . . . . 6  |-  ( H : A -1-1-onto-> B  ->  H : A -onto-> B )
30 forn 5454 . . . . . 6  |-  ( H : A -onto-> B  ->  ran  H  =  B )
3121, 29, 303syl 18 . . . . 5  |-  ( ph  ->  ran  H  =  B )
3231raleqdv 2742 . . . 4  |-  ( ph  ->  ( A. y  e. 
ran  H ( B  i^i  ( `' S " { y } ) )  e.  _V  <->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e.  _V )
)
3328, 32bitr3d 246 . . 3  |-  ( ph  ->  ( A. z  e.  A  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V 
<-> 
A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V ) )
3419, 33sylibd 205 . 2  |-  ( ph  ->  ( R Se  A  ->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V ) )
35 dfse2 5046 . 2  |-  ( S Se  B  <->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V )
3634, 35syl6ibr 218 1  |-  ( ph  ->  ( R Se  A  ->  S Se  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151   {csn 3640   Se wse 4350   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256
This theorem is referenced by:  isose  5840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-se 4353  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264
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