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Theorem isoselem 5581
Description: Lemma for isose 5582. (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 4792 . . . . . . . . 9  |-  ( R Se  A  <->  A. z  e.  A  ( A  i^i  ( `' R " { z } ) )  e. 
_V )
21biimpi 118 . . . . . . . 8  |-  ( R Se  A  ->  A. z  e.  A  ( A  i^i  ( `' R " { z } ) )  e.  _V )
32r19.21bi 2461 . . . . . . 7  |-  ( ( R Se  A  /\  z  e.  A )  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
43expcom 114 . . . . . 6  |-  ( z  e.  A  ->  ( R Se  A  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
)
54adantl 271 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  ( R Se  A  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
)
6 imaeq2 4757 . . . . . . . . . . 11  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( H " x
)  =  ( H
" ( A  i^i  ( `' R " { z } ) ) ) )
76eleq1d 2156 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( ( H "
x )  e.  _V  <->  ( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
87imbi2d 228 . . . . . . . . 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 2679 . . . . . . . 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 270 . . . . . 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 5579 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  z  e.  A )  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
1513, 14sylan 277 . . . . . . 7  |-  ( (
ph  /\  z  e.  A )  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
1615eleq1d 2156 . . . . . 6  |-  ( (
ph  /\  z  e.  A )  ->  (
( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V  <->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
1712, 16sylibd 147 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  (
( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
185, 17syld 44 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  ( R Se  A  ->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e.  _V )
)
1918ralrimdva 2453 . . 3  |-  ( ph  ->  ( R Se  A  ->  A. z  e.  A  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
20 isof1o 5568 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
21 f1ofn 5238 . . . . 5  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
22 sneq 3452 . . . . . . . . 9  |-  ( y  =  ( H `  z )  ->  { y }  =  { ( H `  z ) } )
2322imaeq2d 4761 . . . . . . . 8  |-  ( y  =  ( H `  z )  ->  ( `' S " { y } )  =  ( `' S " { ( H `  z ) } ) )
2423ineq2d 3199 . . . . . . 7  |-  ( y  =  ( H `  z )  ->  ( B  i^i  ( `' S " { y } ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
2524eleq1d 2156 . . . . . 6  |-  ( y  =  ( H `  z )  ->  (
( B  i^i  ( `' S " { y } ) )  e. 
_V 
<->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
2625ralrn 5421 . . . . 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 5244 . . . . . 6  |-  ( H : A -1-1-onto-> B  ->  H : A -onto-> B )
29 forn 5220 . . . . . 6  |-  ( H : A -onto-> B  ->  ran  H  =  B )
3013, 20, 28, 294syl 18 . . . . 5  |-  ( ph  ->  ran  H  =  B )
3130raleqdv 2568 . . . 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 188 . . 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 147 . 2  |-  ( ph  ->  ( R Se  A  ->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V ) )
34 dfse2 4792 . 2  |-  ( S Se  B  <->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V )
3533, 34syl6ibr 160 1  |-  ( ph  ->  ( R Se  A  ->  S Se  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   _Vcvv 2619    i^i cin 2996   {csn 3441   Se wse 4147   `'ccnv 4427   ran crn 4429   "cima 4431    Fn wfn 4997   -onto->wfo 5000   -1-1-onto->wf1o 5001   ` cfv 5002    Isom wiso 5003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-se 4151  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011
This theorem is referenced by:  isose  5582
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