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Theorem isoselem 5673
Description: Lemma for isose 5674. (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 4868 . . . . . . . . 9  |-  ( R Se  A  <->  A. z  e.  A  ( A  i^i  ( `' R " { z } ) )  e. 
_V )
21biimpi 119 . . . . . . . 8  |-  ( R Se  A  ->  A. z  e.  A  ( A  i^i  ( `' R " { z } ) )  e.  _V )
32r19.21bi 2492 . . . . . . 7  |-  ( ( R Se  A  /\  z  e.  A )  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
43expcom 115 . . . . . 6  |-  ( z  e.  A  ->  ( R Se  A  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
)
54adantl 273 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  ( R Se  A  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
)
6 imaeq2 4833 . . . . . . . . . . 11  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( H " x
)  =  ( H
" ( A  i^i  ( `' R " { z } ) ) ) )
76eleq1d 2181 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( ( H "
x )  e.  _V  <->  ( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
87imbi2d 229 . . . . . . . . 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 2715 . . . . . . . 8  |-  ( ( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( ph  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
1110com12 30 . . . . . . 7  |-  ( ph  ->  ( ( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( H "
( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
1211adantr 272 . . . . . 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 5671 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  z  e.  A )  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
1513, 14sylan 279 . . . . . . 7  |-  ( (
ph  /\  z  e.  A )  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
1615eleq1d 2181 . . . . . 6  |-  ( (
ph  /\  z  e.  A )  ->  (
( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V  <->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
1712, 16sylibd 148 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  (
( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
185, 17syld 45 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  ( R Se  A  ->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e.  _V )
)
1918ralrimdva 2484 . . 3  |-  ( ph  ->  ( R Se  A  ->  A. z  e.  A  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
20 isof1o 5660 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
21 f1ofn 5322 . . . . 5  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
22 sneq 3502 . . . . . . . . 9  |-  ( y  =  ( H `  z )  ->  { y }  =  { ( H `  z ) } )
2322imaeq2d 4837 . . . . . . . 8  |-  ( y  =  ( H `  z )  ->  ( `' S " { y } )  =  ( `' S " { ( H `  z ) } ) )
2423ineq2d 3241 . . . . . . 7  |-  ( y  =  ( H `  z )  ->  ( B  i^i  ( `' S " { y } ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
2524eleq1d 2181 . . . . . 6  |-  ( y  =  ( H `  z )  ->  (
( B  i^i  ( `' S " { y } ) )  e. 
_V 
<->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
2625ralrn 5510 . . . . 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 ) )
2713, 20, 21, 264syl 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 )
)
28 f1ofo 5328 . . . . . 6  |-  ( H : A -1-1-onto-> B  ->  H : A -onto-> B )
29 forn 5304 . . . . . 6  |-  ( H : A -onto-> B  ->  ran  H  =  B )
3013, 20, 28, 294syl 18 . . . . 5  |-  ( ph  ->  ran  H  =  B )
3130raleqdv 2604 . . . 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 )
)
3227, 31bitr3d 189 . . 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 ) )
3319, 32sylibd 148 . 2  |-  ( ph  ->  ( R Se  A  ->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V ) )
34 dfse2 4868 . 2  |-  ( S Se  B  <->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V )
3533, 34syl6ibr 161 1  |-  ( ph  ->  ( R Se  A  ->  S Se  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1312    e. wcel 1461   A.wral 2388   _Vcvv 2655    i^i cin 3034   {csn 3491   Se wse 4209   `'ccnv 4496   ran crn 4498   "cima 4500    Fn wfn 5074   -onto->wfo 5077   -1-1-onto->wf1o 5078   ` cfv 5079    Isom wiso 5080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-se 4213  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-isom 5088
This theorem is referenced by:  isose  5674
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