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Theorem isoselem 6236
Description: Lemma for isose 6238. (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 5376 . . . . . . . . 9  |-  ( R Se  A  <->  A. z  e.  A  ( A  i^i  ( `' R " { z } ) )  e. 
_V )
21biimpi 194 . . . . . . . 8  |-  ( R Se  A  ->  A. z  e.  A  ( A  i^i  ( `' R " { z } ) )  e.  _V )
32r19.21bi 2836 . . . . . . 7  |-  ( ( R Se  A  /\  z  e.  A )  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
43expcom 435 . . . . . 6  |-  ( z  e.  A  ->  ( R Se  A  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
)
54adantl 466 . . . . 5  |-  ( (
ph  /\  z  e.  A )  ->  ( R Se  A  ->  ( A  i^i  ( `' R " { z } ) )  e.  _V )
)
6 imaeq2 5339 . . . . . . . . . . 11  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( H " x
)  =  ( H
" ( A  i^i  ( `' R " { z } ) ) ) )
76eleq1d 2536 . . . . . . . . . 10  |-  ( x  =  ( A  i^i  ( `' R " { z } ) )  -> 
( ( H "
x )  e.  _V  <->  ( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
87imbi2d 316 . . . . . . . . 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 3176 . . . . . . . 8  |-  ( ( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( ph  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
1110com12 31 . . . . . . 7  |-  ( ph  ->  ( ( A  i^i  ( `' R " { z } ) )  e. 
_V  ->  ( H "
( A  i^i  ( `' R " { z } ) ) )  e.  _V ) )
1211adantr 465 . . . . . 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 6233 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  z  e.  A )  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
1513, 14sylan 471 . . . . . . 7  |-  ( (
ph  /\  z  e.  A )  ->  ( H " ( A  i^i  ( `' R " { z } ) ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
1615eleq1d 2536 . . . . . 6  |-  ( (
ph  /\  z  e.  A )  ->  (
( H " ( A  i^i  ( `' R " { z } ) ) )  e.  _V  <->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
1712, 16sylibd 214 . . . . 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 2885 . . 3  |-  ( ph  ->  ( R Se  A  ->  A. z  e.  A  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
20 isof1o 6220 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
21 f1ofn 5823 . . . . 5  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
22 sneq 4043 . . . . . . . . 9  |-  ( y  =  ( H `  z )  ->  { y }  =  { ( H `  z ) } )
2322imaeq2d 5343 . . . . . . . 8  |-  ( y  =  ( H `  z )  ->  ( `' S " { y } )  =  ( `' S " { ( H `  z ) } ) )
2423ineq2d 3705 . . . . . . 7  |-  ( y  =  ( H `  z )  ->  ( B  i^i  ( `' S " { y } ) )  =  ( B  i^i  ( `' S " { ( H `  z ) } ) ) )
2524eleq1d 2536 . . . . . 6  |-  ( y  =  ( H `  z )  ->  (
( B  i^i  ( `' S " { y } ) )  e. 
_V 
<->  ( B  i^i  ( `' S " { ( H `  z ) } ) )  e. 
_V ) )
2625ralrn 6035 . . . . 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 21 . . . 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 5829 . . . . . 6  |-  ( H : A -1-1-onto-> B  ->  H : A -onto-> B )
29 forn 5804 . . . . . 6  |-  ( H : A -onto-> B  ->  ran  H  =  B )
3013, 20, 28, 294syl 21 . . . . 5  |-  ( ph  ->  ran  H  =  B )
3130raleqdv 3069 . . . 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 255 . . 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 214 . 2  |-  ( ph  ->  ( R Se  A  ->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V ) )
34 dfse2 5376 . 2  |-  ( S Se  B  <->  A. y  e.  B  ( B  i^i  ( `' S " { y } ) )  e. 
_V )
3533, 34syl6ibr 227 1  |-  ( ph  ->  ( R Se  A  ->  S Se  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    i^i cin 3480   {csn 4033   Se wse 4842   `'ccnv 5004   ran crn 5006   "cima 5008    Fn wfn 5589   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-se 4845  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603
This theorem is referenced by:  isose  6238
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