Theorem nfiso 5446

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 4911	. 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 5124	. . 3 |- F/ x H : A -1-1-onto-> B
6		nfcv 2178	. . . . . . 7 |- F/_ x y
7		nfiso.2	. . . . . . 7 |- F/_ x R
8		nfcv 2178	. . . . . . 7 |- F/_ x z
9	6, 7, 8	nfbr 3808	. . . . . 6 |- F/ x y R z
10	2, 6	nffv 5185	. . . . . . 7 |- F/_ x ( H ` y )
11		nfiso.3	. . . . . . 7 |- F/_ x S
12	2, 8	nffv 5185	. . . . . . 7 |- F/_ x ( H ` z )
13	10, 11, 12	nfbr 3808	. . . . . 6 |- F/ x ( H ` y ) S ( H ` z )
14	9, 13	nfbi 1481	. . . . 5 |- F/ x ( y R z <-> ( H ` y ) S ( H ` z ) )
15	3, 14	nfralxy 2360	. . . 4 |- F/ x A. z e. A ( y R z <-> ( H ` y ) S ( H ` z ) )
16	3, 15	nfralxy 2360	. . 3 |- F/ x A. y e. A A. z e. A ( y R z <-> ( H ` y ) S ( H ` z ) )
17	5, 16	nfan 1457	. 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 1363	1 |- F/ x H Isom R , S ( A , B )

Syntax hints:   /\ wa 97   <-> wb 98   F/wnf 1349   F/_wnfc 2165   A.wral 2306   class class class wbr 3764   -1-1-onto->wf1o 4901   ` cfv 4902   Isom wiso 4903

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-isom 4911

This theorem is referenced by: (None)