Theorem nfiso 5978

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 5360	. 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 5611	. . 3 ⊢ Ⅎ𝑥 𝐻:𝐴–1-1-onto→𝐵
6		nfcv 2384	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
7		nfiso.2	. . . . . . 7 ⊢ Ⅎ𝑥𝑅
8		nfcv 2384	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
9	6, 7, 8	nfbr 4155	. . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧
10	2, 6	nffv 5679	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦)
11		nfiso.3	. . . . . . 7 ⊢ Ⅎ𝑥𝑆
12	2, 8	nffv 5679	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧)
13	10, 11, 12	nfbr 4155	. . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧)
14	9, 13	nfbi 1638	. . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
15	3, 14	nfralxy 2580	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
16	3, 15	nfralxy 2580	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))
17	5, 16	nfan 1614	. 2 ⊢ Ⅎ𝑥(𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)))
18	1, 17	nfxfr 1523	1 ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)

Syntax hints:   ∧ wa 104   ↔ wb 105  Ⅎwnf 1509  Ⅎwnfc 2371  ∀wral 2520   class class class wbr 4108  –1-1-onto→wf1o 5350  ‘cfv 5351   Isom wiso 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360

This theorem is referenced by: (None)