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Theorem isofr2 5857
Description: A weak form of isofr 5855 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
Assertion
Ref Expression
isofr2  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  B  e.  V )  ->  ( S  Fr  B  ->  R  Fr  A ) )

Proof of Theorem isofr2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . 2  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  B  e.  V )  ->  H  Isom  R ,  S  ( A ,  B ) )
2 imassrn 5041 . . . 4  |-  ( H
" x )  C_  ran  H
3 isof1o 5838 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
4 f1of 5488 . . . . 5  |-  ( H : A -1-1-onto-> B  ->  H : A
--> B )
5 frn 5411 . . . . 5  |-  ( H : A --> B  ->  ran  H  C_  B )
63, 4, 53syl 18 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ran  H  C_  B
)
72, 6syl5ss 3203 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  C_  B
)
8 ssexg 4176 . . 3  |-  ( ( ( H " x
)  C_  B  /\  B  e.  V )  ->  ( H " x
)  e.  _V )
97, 8sylan 457 . 2  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  B  e.  V )  ->  ( H " x )  e. 
_V )
101, 9isofrlem 5853 1  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  B  e.  V )  ->  ( S  Fr  B  ->  R  Fr  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   _Vcvv 2801    C_ wss 3165    Fr wfr 4365   ran crn 4706   "cima 4708   -->wf 5267   -1-1-onto->wf1o 5270    Isom wiso 5272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-fr 4368  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280
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