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Theorem isofr2 5841
Description: A weak form of isofr 5839 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 5025 . . . 4  |-  ( H
" x )  C_  ran  H
3 isof1o 5822 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
4 f1of 5472 . . . . 5  |-  ( H : A -1-1-onto-> B  ->  H : A
--> B )
5 frn 5395 . . . . 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 3190 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  C_  B
)
8 ssexg 4160 . . 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 5837 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 1684   _Vcvv 2788    C_ wss 3152    Fr wfr 4349   ran crn 4690   "cima 4692   -->wf 5251   -1-1-onto->wf1o 5254    Isom wiso 5256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-fr 4352  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264
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