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Theorem nfiso 5821
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 5264 . 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 5470 . . 3  |-  F/ x  H : A -1-1-onto-> B
6 nfcv 2419 . . . . . . 7  |-  F/_ x
y
7 nfiso.2 . . . . . . 7  |-  F/_ x R
8 nfcv 2419 . . . . . . 7  |-  F/_ x
z
96, 7, 8nfbr 4067 . . . . . 6  |-  F/ x  y R z
102, 6nffv 5532 . . . . . . 7  |-  F/_ x
( H `  y
)
11 nfiso.3 . . . . . . 7  |-  F/_ x S
122, 8nffv 5532 . . . . . . 7  |-  F/_ x
( H `  z
)
1310, 11, 12nfbr 4067 . . . . . 6  |-  F/ x
( H `  y
) S ( H `
 z )
149, 13nfbi 1772 . . . . 5  |-  F/ x
( y R z  <-> 
( H `  y
) S ( H `
 z ) )
153, 14nfral 2596 . . . 4  |-  F/ x A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) )
163, 15nfral 2596 . . 3  |-  F/ x A. y  e.  A  A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) )
175, 16nfan 1771 . 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 1557 1  |-  F/ x  H  Isom  R ,  S  ( A ,  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   F/wnf 1531   F/_wnfc 2406   A.wral 2543   class class class wbr 4023   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264
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