Theorem nfiso 5952

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 5337	. 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 5584	. . 3 ⊢ Ⅎ𝑥 𝐻:𝐴–1-1-onto→𝐵
6		nfcv 2373	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
7		nfiso.2	. . . . . . 7 ⊢ Ⅎ𝑥𝑅
8		nfcv 2373	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
9	6, 7, 8	nfbr 4136	. . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧
10	2, 6	nffv 5652	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦)
11		nfiso.3	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
12	2, 8	nffv 5652	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧)
13	10, 11, 12	nfbr 4136	. . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧)
14	9, 13	nfbi 1637	. . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
15	3, 14	nfralxy 2569	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
16	3, 15	nfralxy 2569	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
17	5, 16	nfan 1613	. 2 ⊢ Ⅎ𝑥(𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)))
18	1, 17	nfxfr 1522	1 ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)

Syntax hints:   ∧ wa 104   ↔ wb 105  Ⅎwnf 1508  Ⅎwnfc 2360  ∀wral 2509   class class class wbr 4089  –1-1-onto→wf1o 5327  ‘cfv 5328   Isom wiso 5329

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-isom 5337

This theorem is referenced by: (None)