Theorem nfiso 7343

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 6569	. 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 6845	. . 3 ⊢ Ⅎ𝑥 𝐻:𝐴–1-1-onto→𝐵
6		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
7		nfiso.2	. . . . . . 7 ⊢ Ⅎ𝑥𝑅
8		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
9	6, 7, 8	nfbr 5189	. . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧
10	2, 6	nffv 6915	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦)
11		nfiso.3	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
12	2, 8	nffv 6915	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧)
13	10, 11, 12	nfbr 5189	. . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧)
14	9, 13	nfbi 1902	. . . . 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 1898	. 2 ⊢ Ⅎ𝑥(𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)))
18	1, 17	nfxfr 1852	1 ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395  Ⅎwnf 1782  Ⅎwnfc 2889  ∀wral 3060   class class class wbr 5142  –1-1-onto→wf1o 6559  ‘cfv 6560   Isom wiso 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569

This theorem is referenced by: (None)