Theorem nfiso 7054

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	⊢ Ⅎ𝑥𝐻
nfiso.2	⊢ Ⅎ𝑥𝑅
nfiso.3	⊢ Ⅎ𝑥𝑆
nfiso.4	⊢ Ⅎ𝑥𝐴
nfiso.5	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfiso	⊢ Ⅎ𝑥 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 6333	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))))
2		nfiso.1	. . . 4 ⊢ Ⅎ𝑥𝐻
3		nfiso.4	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfiso.5	. . . 4 ⊢ Ⅎ𝑥𝐵
5	2, 3, 4	nff1o 6588	. . 3 ⊢ Ⅎ𝑥 𝐻:𝐴–1-1-onto→𝐵
6		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
7		nfiso.2	. . . . . . 7 ⊢ Ⅎ𝑥𝑅
8		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
9	6, 7, 8	nfbr 5077	. . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧
10	2, 6	nffv 6655	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦)
11		nfiso.3	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
12	2, 8	nffv 6655	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧)
13	10, 11, 12	nfbr 5077	. . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧)
14	9, 13	nfbi 1904	. . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
15	3, 14	nfralw 3189	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
16	3, 15	nfralw 3189	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
17	5, 16	nfan 1900	. 2 ⊢ Ⅎ𝑥(𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)))
18	1, 17	nfxfr 1854	1 ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)

Syntax hints:   ↔ wb 209   ∧ wa 399  Ⅎwnf 1785  Ⅎwnfc 2936  ∀wral 3106   class class class wbr 5030  –1-1-onto→wf1o 6323  ‘cfv 6324   Isom wiso 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333

This theorem is referenced by: (None)