Theorem nfiso 5823

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.

Step	Hyp	Ref	Expression
1		df-isom 5240	. 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
5	2, 3, 4	nff1o 5474	. . 3 |- F/ x H : A -1-1-onto-> B
6		nfcv 2332	. . . . . . 7 |- F/_ x y
7		nfiso.2	. . . . . . 7 |- F/_ x R
8		nfcv 2332	. . . . . . 7 |- F/_ x z
9	6, 7, 8	nfbr 4064	. . . . . 6 |- F/ x y R z
10	2, 6	nffv 5540	. . . . . . 7 |- F/_ x ( H ` y )
11		nfiso.3	. . . . . . 7 |- F/_ x S
12	2, 8	nffv 5540	. . . . . . 7 |- F/_ x ( H ` z )
13	10, 11, 12	nfbr 4064	. . . . . 6 |- F/ x ( H ` y ) S ( H ` z )
14	9, 13	nfbi 1600	. . . . 5 |- F/ x ( y R z <-> ( H ` y ) S ( H ` z ) )
15	3, 14	nfralxy 2528	. . . 4 |- F/ x A. z e. A ( y R z <-> ( H ` y ) S ( H ` z ) )
16	3, 15	nfralxy 2528	. . 3 |- F/ x A. y e. A A. z e. A ( y R z <-> ( H ` y ) S ( H ` z ) )
17	5, 16	nfan 1576	. 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 ) ) )
18	1, 17	nfxfr 1485	1 |- F/ x H Isom R , S ( A , B )

Syntax hints:   /\ wa 104   <-> wb 105   F/wnf 1471   F/_wnfc 2319   A.wral 2468   class class class wbr 4018   -1-1-onto->wf1o 5230   ` cfv 5231   Isom wiso 5232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240

This theorem is referenced by: (None)