Theorem nfiso 5898

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 5299	. 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 5542	. . 3 |- F/ x H : A -1-1-onto-> B
6		nfcv 2350	. . . . . . 7 |- F/_ x y
7		nfiso.2	. . . . . . 7 |- F/_ x R
8		nfcv 2350	. . . . . . 7 |- F/_ x z
9	6, 7, 8	nfbr 4106	. . . . . 6 |- F/ x y R z
10	2, 6	nffv 5609	. . . . . . 7 |- F/_ x ( H ` y )
11		nfiso.3	. . . . . . 7 |- F/_ x S
12	2, 8	nffv 5609	. . . . . . 7 |- F/_ x ( H ` z )
13	10, 11, 12	nfbr 4106	. . . . . 6 |- F/ x ( H ` y ) S ( H ` z )
14	9, 13	nfbi 1613	. . . . 5 |- F/ x ( y R z <-> ( H ` y ) S ( H ` z ) )
15	3, 14	nfralxy 2546	. . . 4 |- F/ x A. z e. A ( y R z <-> ( H ` y ) S ( H ` z ) )
16	3, 15	nfralxy 2546	. . 3 |- F/ x A. y e. A A. z e. A ( y R z <-> ( H ` y ) S ( H ` z ) )
17	5, 16	nfan 1589	. 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 1498	1 |- F/ x H Isom R , S ( A , B )

Syntax hints:   /\ wa 104   <-> wb 105   F/wnf 1484   F/_wnfc 2337   A.wral 2486   class class class wbr 4059   -1-1-onto->wf1o 5289   ` cfv 5290   Isom wiso 5291

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299

This theorem is referenced by: (None)