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.

Step	Hyp	Ref	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
5	2, 3, 4	nff1o 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
9	6, 7, 8	nfbr 4440	. . . . . 6 |- F/ x y R z
10	2, 6	nffv 5886	. . . . . . 7 |- F/_ x ( H ` y )
11		nfiso.3	. . . . . . 7 |- F/_ x S
12	2, 8	nffv 5886	. . . . . . 7 |- F/_ x ( H ` z )
13	10, 11, 12	nfbr 4440	. . . . . 6 |- F/ x ( H ` y ) S ( H ` z )
14	9, 13	nfbi 2037	. . . . 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 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 ) ) )
18	1, 17	nfxfr 1704	1 |- F/ x H Isom R , S ( A , B )

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)