Theorem nfiso 5488

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-iso 4797	. 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 5286	. . 3 ⊢ Ⅎx H:A–1-1-onto→B
6		nfcv 2490	. . . . . . 7 ⊢ Ⅎxy
7		nfiso.2	. . . . . . 7 ⊢ ℲxR
8		nfcv 2490	. . . . . . 7 ⊢ Ⅎxz
9	6, 7, 8	nfbr 4684	. . . . . 6 ⊢ Ⅎx yRz
10	2, 6	nffv 5335	. . . . . . 7 ⊢ Ⅎx(H ‘y)
11		nfiso.3	. . . . . . 7 ⊢ ℲxS
12	2, 8	nffv 5335	. . . . . . 7 ⊢ Ⅎx(H ‘z)
13	10, 11, 12	nfbr 4684	. . . . . 6 ⊢ Ⅎx(H ‘y)S(H ‘z)
14	9, 13	nfbi 1834	. . . . 5 ⊢ Ⅎx(yRz ↔ (H ‘y)S(H ‘z))
15	3, 14	nfral 2668	. . . 4 ⊢ Ⅎx∀z ∈ A (yRz ↔ (H ‘y)S(H ‘z))
16	3, 15	nfral 2668	. . 3 ⊢ Ⅎx∀y ∈ A ∀z ∈ A (yRz ↔ (H ‘y)S(H ‘z))
17	5, 16	nfan 1824	. 2 ⊢ Ⅎx(H:A–1-1-onto→B ∧ ∀y ∈ A ∀z ∈ A (yRz ↔ (H ‘y)S(H ‘z)))
18	1, 17	nfxfr 1570	1 ⊢ Ⅎx H Isom R, S (A, B)

Syntax hints:   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544  Ⅎwnfc 2477  ∀wral 2615   class class class wbr 4640  –1-1-onto→wf1o 4781   ‘cfv 4782   Isom wiso 4783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-iso 4797

This theorem is referenced by: (None)