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Theorem nfiso 5837
Description: Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
nfiso.1  |-  F/_ x H
nfiso.2  |-  F/_ x R
nfiso.3  |-  F/_ x S
nfiso.4  |-  F/_ x A
nfiso.5  |-  F/_ x B
Assertion
Ref Expression
nfiso  |-  F/ x  H  Isom  R ,  S  ( A ,  B )

Proof of Theorem nfiso
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-isom 5280 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
( H : A -1-1-onto-> B  /\  A. y  e.  A  A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) ) ) )
2 nfiso.1 . . . 4  |-  F/_ x H
3 nfiso.4 . . . 4  |-  F/_ x A
4 nfiso.5 . . . 4  |-  F/_ x B
52, 3, 4nff1o 5486 . . 3  |-  F/ x  H : A -1-1-onto-> B
6 nfcv 2432 . . . . . . 7  |-  F/_ x
y
7 nfiso.2 . . . . . . 7  |-  F/_ x R
8 nfcv 2432 . . . . . . 7  |-  F/_ x
z
96, 7, 8nfbr 4083 . . . . . 6  |-  F/ x  y R z
102, 6nffv 5548 . . . . . . 7  |-  F/_ x
( H `  y
)
11 nfiso.3 . . . . . . 7  |-  F/_ x S
122, 8nffv 5548 . . . . . . 7  |-  F/_ x
( H `  z
)
1310, 11, 12nfbr 4083 . . . . . 6  |-  F/ x
( H `  y
) S ( H `
 z )
149, 13nfbi 1784 . . . . 5  |-  F/ x
( y R z  <-> 
( H `  y
) S ( H `
 z ) )
153, 14nfral 2609 . . . 4  |-  F/ x A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) )
163, 15nfral 2609 . . 3  |-  F/ x A. y  e.  A  A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) )
175, 16nfan 1783 . 2  |-  F/ x
( H : A -1-1-onto-> B  /\  A. y  e.  A  A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) ) )
181, 17nfxfr 1560 1  |-  F/ x  H  Isom  R ,  S  ( A ,  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   F/wnf 1534   F/_wnfc 2419   A.wral 2556   class class class wbr 4039   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  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-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  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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