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Theorem isowe2 5847
Description: A weak form of isowe 5846 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
Assertion
Ref Expression
isowe2  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  We  B  ->  R  We  A
) )
Distinct variable groups:    x, A    x, B    x, R    x, S    x, H

Proof of Theorem isowe2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . . 4  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  H  Isom  R ,  S  ( A ,  B ) )
2 imaeq2 5008 . . . . . . 7  |-  ( x  =  y  ->  ( H " x )  =  ( H " y
) )
32eleq1d 2349 . . . . . 6  |-  ( x  =  y  ->  (
( H " x
)  e.  _V  <->  ( H " y )  e.  _V ) )
43spv 1938 . . . . 5  |-  ( A. x ( H "
x )  e.  _V  ->  ( H " y
)  e.  _V )
54adantl 452 . . . 4  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( H "
y )  e.  _V )
61, 5isofrlem 5837 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  Fr  B  ->  R  Fr  A
) )
7 isosolem 5844 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A
) )
87adantr 451 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  Or  B  ->  R  Or  A
) )
96, 8anim12d 546 . 2  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( ( S  Fr  B  /\  S  Or  B )  ->  ( R  Fr  A  /\  R  Or  A )
) )
10 df-we 4354 . 2  |-  ( S  We  B  <->  ( S  Fr  B  /\  S  Or  B ) )
11 df-we 4354 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
129, 10, 113imtr4g 261 1  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  We  B  ->  R  We  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788    Or wor 4313    Fr wfr 4349    We wwe 4351   "cima 4692    Isom wiso 5256
This theorem is referenced by:  fnwelem  6230  ltweuz  11024
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-3or 935  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-po 4314  df-so 4315  df-fr 4352  df-we 4354  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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