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Theorem nfiso 6233
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 5598 . 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 5826 . . 3  |-  F/ x  H : A -1-1-onto-> B
6 nfcv 2612 . . . . . . 7  |-  F/_ x
y
7 nfiso.2 . . . . . . 7  |-  F/_ x R
8 nfcv 2612 . . . . . . 7  |-  F/_ x
z
96, 7, 8nfbr 4440 . . . . . 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 4440 . . . . . 6  |-  F/ x
( H `  y
) S ( H `
 z )
149, 13nfbi 2037 . . . . 5  |-  F/ x
( y R z  <-> 
( H `  y
) S ( H `
 z ) )
153, 14nfral 2789 . . . 4  |-  F/ x A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) )
163, 15nfral 2789 . . 3  |-  F/ x A. y  e.  A  A. z  e.  A  ( y R z  <-> 
( H `  y
) S ( H `
 z ) )
175, 16nfan 2031 . 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 1704 1  |-  F/ x  H  Isom  R ,  S  ( A ,  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376   F/wnf 1675   F/_wnfc 2599   A.wral 2756   class class class wbr 4395   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598
This theorem is referenced by: (None)
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