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Theorem isowe2 6232
Description: A weak form of isowe 6231 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 457 . . . 4  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  H  Isom  R ,  S  ( A ,  B ) )
2 imaeq2 5331 . . . . . . 7  |-  ( x  =  y  ->  ( H " x )  =  ( H " y
) )
32eleq1d 2536 . . . . . 6  |-  ( x  =  y  ->  (
( H " x
)  e.  _V  <->  ( H " y )  e.  _V ) )
43spv 1980 . . . . 5  |-  ( A. x ( H "
x )  e.  _V  ->  ( H " y
)  e.  _V )
54adantl 466 . . . 4  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( H "
y )  e.  _V )
61, 5isofrlem 6222 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  Fr  B  ->  R  Fr  A
) )
7 isosolem 6229 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A
) )
87adantr 465 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  Or  B  ->  R  Or  A
) )
96, 8anim12d 563 . 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 4840 . 2  |-  ( S  We  B  <->  ( S  Fr  B  /\  S  Or  B ) )
11 df-we 4840 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
129, 10, 113imtr4g 270 1  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  We  B  ->  R  We  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377    e. wcel 1767   _Vcvv 3113    Or wor 4799    Fr wfr 4835    We wwe 4837   "cima 5002    Isom wiso 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595
This theorem is referenced by:  fnwelem  6895  ltweuz  12036
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