Theorem nfiso 6202

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 5577	. 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 5796	. . 3 |- F/ x H : A -1-1-onto-> B
6		nfcv 2564	. . . . . . 7 |- F/_ x y
7		nfiso.2	. . . . . . 7 |- F/_ x R
8		nfcv 2564	. . . . . . 7 |- F/_ x z
9	6, 7, 8	nfbr 4438	. . . . . 6 |- F/ x y R z
10	2, 6	nffv 5855	. . . . . . 7 |- F/_ x ( H ` y )
11		nfiso.3	. . . . . . 7 |- F/_ x S
12	2, 8	nffv 5855	. . . . . . 7 |- F/_ x ( H ` z )
13	10, 11, 12	nfbr 4438	. . . . . 6 |- F/ x ( H ` y ) S ( H ` z )
14	9, 13	nfbi 1962	. . . . 5 |- F/ x ( y R z <-> ( H ` y ) S ( H ` z ) )
15	3, 14	nfral 2789	. . . 4 |- F/ x A. z e. A ( y R z <-> ( H ` y ) S ( H ` z ) )
16	3, 15	nfral 2789	. . 3 |- F/ x A. y e. A A. z e. A ( y R z <-> ( H ` y ) S ( H ` z ) )
17	5, 16	nfan 1956	. 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 1666	1 |- F/ x H Isom R , S ( A , B )

Syntax hints:   <-> wb 184   /\ wa 367   F/wnf 1637   F/_wnfc 2550   A.wral 2753   class class class wbr 4394   -1-1-onto->wf1o 5567   ` cfv 5568   Isom wiso 5569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577

This theorem is referenced by: (None)