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Theorem nfiso 6228
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 5608 . 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 5827 . . 3  |-  F/ x  H : A -1-1-onto-> B
6 nfcv 2585 . . . . . . 7  |-  F/_ x
y
7 nfiso.2 . . . . . . 7  |-  F/_ x R
8 nfcv 2585 . . . . . . 7  |-  F/_ x
z
96, 7, 8nfbr 4466 . . . . . 6  |-  F/ x  y R z
102, 6nffv 5886 . . . . . . 7  |-  F/_ x
( H `  y
)
11 nfiso.3 . . . . . . 7  |-  F/_ x S
122, 8nffv 5886 . . . . . . 7  |-  F/_ x
( H `  z
)
1310, 11, 12nfbr 4466 . . . . . 6  |-  F/ x
( H `  y
) S ( H `
 z )
149, 13nfbi 1991 . . . . 5  |-  F/ x
( y R z  <-> 
( H `  y
) S ( H `
 z ) )
153, 14nfral 2812 . . . 4  |-  F/ x A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) )
163, 15nfral 2812 . . 3  |-  F/ x A. y  e.  A  A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) )
175, 16nfan 1985 . 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 1693 1  |-  F/ x  H  Isom  R ,  S  ( A ,  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371   F/wnf 1664   F/_wnfc 2571   A.wral 2776   class class class wbr 4421   -1-1-onto->wf1o 5598   ` cfv 5599    Isom wiso 5600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608
This theorem is referenced by: (None)
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