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Theorem isoselem 6053
Description: Lemma for isose 6055. (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 5229 . . . . . . . . 9  |-  ( R Se  A  <->  A. z  e.  A  ( A  i^i  ( `' R " { z } ) )  e. 
_V )
21biimpi 187 . . . . . . . 8  |-  ( R Se  A  ->  A. z  e.  A  ( A  i^i  ( `' R " { z } ) )  e.  _V )
32r19.21bi 2796 . . . . . . 7  |-  ( ( R Se  A  /\  z  e.  A )  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
43expcom 425 . . . . . 6  |-  ( z  e.  A  ->  ( R Se  A  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
)
54adantl 453 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  ( R Se  A  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
)
6 imaeq2 5191 . . . . . . . . . . 11  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( H " x
)  =  ( H
" ( A  i^i  ( `' R " { z } ) ) ) )
76eleq1d 2501 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( ( H "
x )  e.  _V  <->  ( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
87imbi2d 308 . . . . . . . . 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 3003 . . . . . . . 8  |-  ( ( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( ph  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
1110com12 29 . . . . . . 7  |-  ( ph  ->  ( ( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( H "
( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
1211adantr 452 . . . . . 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 6050 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  z  e.  A )  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
1513, 14sylan 458 . . . . . . 7  |-  ( (
ph  /\  z  e.  A )  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
1615eleq1d 2501 . . . . . 6  |-  ( (
ph  /\  z  e.  A )  ->  (
( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V  <->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
1712, 16sylibd 206 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  (
( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
185, 17syld 42 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  ( R Se  A  ->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e.  _V )
)
1918ralrimdva 2788 . . 3  |-  ( ph  ->  ( R Se  A  ->  A. z  e.  A  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
20 isof1o 6037 . . . . . 6  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
2113, 20syl 16 . . . . 5  |-  ( ph  ->  H : A -1-1-onto-> B )
22 f1ofn 5667 . . . . 5  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
23 sneq 3817 . . . . . . . . 9  |-  ( y  =  ( H `  z )  ->  { y }  =  { ( H `  z ) } )
2423imaeq2d 5195 . . . . . . . 8  |-  ( y  =  ( H `  z )  ->  ( `' S " { y } )  =  ( `' S " { ( H `  z ) } ) )
2524ineq2d 3534 . . . . . . 7  |-  ( y  =  ( H `  z )  ->  ( B  i^i  ( `' S " { y } ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
2625eleq1d 2501 . . . . . 6  |-  ( y  =  ( H `  z )  ->  (
( B  i^i  ( `' S " { y } ) )  e. 
_V 
<->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
2726ralrn 5865 . . . . 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 19 . . . 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 5673 . . . . . 6  |-  ( H : A -1-1-onto-> B  ->  H : A -onto-> B )
30 forn 5648 . . . . . 6  |-  ( H : A -onto-> B  ->  ran  H  =  B )
3121, 29, 303syl 19 . . . . 5  |-  ( ph  ->  ran  H  =  B )
3231raleqdv 2902 . . . 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 247 . . 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 206 . 2  |-  ( ph  ->  ( R Se  A  ->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V ) )
35 dfse2 5229 . 2  |-  ( S Se  B  <->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V )
3634, 35syl6ibr 219 1  |-  ( ph  ->  ( R Se  A  ->  S Se  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311   {csn 3806   Se wse 4531   `'ccnv 4869   ran crn 4871   "cima 4873    Fn wfn 5441   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446    Isom wiso 5447
This theorem is referenced by:  isose  6055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-se 4534  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455
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