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Theorem nfiso 6199
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 5588 . 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 5805 . . 3  |-  F/ x  H : A -1-1-onto-> B
6 nfcv 2622 . . . . . . 7  |-  F/_ x
y
7 nfiso.2 . . . . . . 7  |-  F/_ x R
8 nfcv 2622 . . . . . . 7  |-  F/_ x
z
96, 7, 8nfbr 4484 . . . . . 6  |-  F/ x  y R z
102, 6nffv 5864 . . . . . . 7  |-  F/_ x
( H `  y
)
11 nfiso.3 . . . . . . 7  |-  F/_ x S
122, 8nffv 5864 . . . . . . 7  |-  F/_ x
( H `  z
)
1310, 11, 12nfbr 4484 . . . . . 6  |-  F/ x
( H `  y
) S ( H `
 z )
149, 13nfbi 1876 . . . . 5  |-  F/ x
( y R z  <-> 
( H `  y
) S ( H `
 z ) )
153, 14nfral 2843 . . . 4  |-  F/ x A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) )
163, 15nfral 2843 . . 3  |-  F/ x A. y  e.  A  A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) )
175, 16nfan 1870 . 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 1620 1  |-  F/ x  H  Isom  R ,  S  ( A ,  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   F/wnf 1594   F/_wnfc 2608   A.wral 2807   class class class wbr 4440   -1-1-onto->wf1o 5578   ` cfv 5579    Isom wiso 5580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588
This theorem is referenced by: (None)
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