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Theorem isowe2 6142
Description: A weak form of isowe 6141 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 5265 . . . . . . 7  |-  ( x  =  y  ->  ( H " x )  =  ( H " y
) )
32eleq1d 2520 . . . . . 6  |-  ( x  =  y  ->  (
( H " x
)  e.  _V  <->  ( H " y )  e.  _V ) )
43spv 1964 . . . . 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 6132 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H "
x )  e.  _V )  ->  ( S  Fr  B  ->  R  Fr  A
) )
7 isosolem 6139 . . . 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 4781 . 2  |-  ( S  We  B  <->  ( S  Fr  B  /\  S  Or  B ) )
11 df-we 4781 . 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 1368    e. wcel 1758   _Vcvv 3070    Or wor 4740    Fr wfr 4776    We wwe 4778   "cima 4943    Isom wiso 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527
This theorem is referenced by:  fnwelem  6789  ltweuz  11887
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