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Theorem isowe2 6009
Description: A weak form of isowe 6008 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 444 . . . 4  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  H  Isom  R ,  S  ( A ,  B ) )
2 imaeq2 5139 . . . . . . 7  |-  ( x  =  y  ->  ( H " x )  =  ( H " y
) )
32eleq1d 2453 . . . . . 6  |-  ( x  =  y  ->  (
( H " x
)  e.  _V  <->  ( H " y )  e.  _V ) )
43spv 2032 . . . . 5  |-  ( A. x ( H "
x )  e.  _V  ->  ( H " y
)  e.  _V )
54adantl 453 . . . 4  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( H "
y )  e.  _V )
61, 5isofrlem 5999 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  Fr  B  ->  R  Fr  A
) )
7 isosolem 6006 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A
) )
87adantr 452 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  Or  B  ->  R  Or  A
) )
96, 8anim12d 547 . 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 4484 . 2  |-  ( S  We  B  <->  ( S  Fr  B  /\  S  Or  B ) )
11 df-we 4484 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
129, 10, 113imtr4g 262 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 359   A.wal 1546    e. wcel 1717   _Vcvv 2899    Or wor 4443    Fr wfr 4479    We wwe 4481   "cima 4821    Isom wiso 5395
This theorem is referenced by:  fnwelem  6397  ltweuz  11228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403
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