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Theorem isofr2 6064
Description: A weak form of isofr 6062 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 444 . 2  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  B  e.  V )  ->  H  Isom  R ,  S  ( A ,  B ) )
2 imassrn 5216 . . . 4  |-  ( H
" x )  C_  ran  H
3 isof1o 6045 . . . . 5  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
4 f1of 5674 . . . . 5  |-  ( H : A -1-1-onto-> B  ->  H : A
--> B )
5 frn 5597 . . . . 5  |-  ( H : A --> B  ->  ran  H  C_  B )
63, 4, 53syl 19 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ran  H  C_  B
)
72, 6syl5ss 3359 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  C_  B
)
8 ssexg 4349 . . 3  |-  ( ( ( H " x
)  C_  B  /\  B  e.  V )  ->  ( H " x
)  e.  _V )
97, 8sylan 458 . 2  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  B  e.  V )  ->  ( H " x )  e. 
_V )
101, 9isofrlem 6060 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 359    e. wcel 1725   _Vcvv 2956    C_ wss 3320    Fr wfr 4538   ran crn 4879   "cima 4881   -->wf 5450   -1-1-onto->wf1o 5453    Isom wiso 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-fr 4541  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463
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