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Theorem isowe2 5863
Description: A weak form of isowe 5862 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 5024 . . . . . . 7  |-  ( x  =  y  ->  ( H " x )  =  ( H " y
) )
32eleq1d 2362 . . . . . 6  |-  ( x  =  y  ->  (
( H " x
)  e.  _V  <->  ( H " y )  e.  _V ) )
43spv 1951 . . . . 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 5853 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  Fr  B  ->  R  Fr  A
) )
7 isosolem 5860 . . . 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 4370 . 2  |-  ( S  We  B  <->  ( S  Fr  B  /\  S  Or  B ) )
11 df-we 4370 . 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 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801    Or wor 4329    Fr wfr 4365    We wwe 4367   "cima 4708    Isom wiso 5272
This theorem is referenced by:  fnwelem  6246  ltweuz  11040
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-3or 935  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-po 4330  df-so 4331  df-fr 4368  df-we 4370  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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