Theorem nfiso 7320

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 6545	. 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 6818	. . 3 ⊢ Ⅎ𝑥 𝐻:𝐴–1-1-onto→𝐵
6		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
7		nfiso.2	. . . . . . 7 ⊢ Ⅎ𝑥𝑅
8		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
9	6, 7, 8	nfbr 5157	. . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧
10	2, 6	nffv 6891	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦)
11		nfiso.3	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
12	2, 8	nffv 6891	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧)
13	10, 11, 12	nfbr 5157	. . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧)
14	9, 13	nfbi 1931	. . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
15	3, 14	nfralw 3310	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
16	3, 15	nfralw 3310	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
17	5, 16	nfan 1927	. 2 ⊢ Ⅎ𝑥(𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)))
18	1, 17	nfxfr 1881	1 ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)

Syntax hints:   ↔ wb 209   ∧ wa 400  Ⅎwnf 1811  Ⅎwnfc 2908  ∀wral 3077   class class class wbr 5108  –1-1-onto→wf1o 6535  ‘cfv 6536   Isom wiso 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545

This theorem is referenced by: (None)