Theorem nfiso 5487

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 4796	. 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 5285	. . 3 ⊢ Ⅎx H:A–1-1-onto→B
6		nfcv 2489	. . . . . . 7 ⊢ Ⅎxy
7		nfiso.2	. . . . . . 7 ⊢ ℲxR
8		nfcv 2489	. . . . . . 7 ⊢ Ⅎxz
9	6, 7, 8	nfbr 4683	. . . . . 6 ⊢ Ⅎx yRz
10	2, 6	nffv 5334	. . . . . . 7 ⊢ Ⅎx(H ‘y)
11		nfiso.3	. . . . . . 7 ⊢ ℲxS
12	2, 8	nffv 5334	. . . . . . 7 ⊢ Ⅎx(H ‘z)
13	10, 11, 12	nfbr 4683	. . . . . 6 ⊢ Ⅎx(H ‘y)S(H ‘z)
14	9, 13	nfbi 1834	. . . . 5 ⊢ Ⅎx(yRz ↔ (H ‘y)S(H ‘z))
15	3, 14	nfral 2667	. . . 4 ⊢ Ⅎx∀z ∈ A (yRz ↔ (H ‘y)S(H ‘z))
16	3, 15	nfral 2667	. . 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 2476  ∀wral 2614   class class class wbr 4639  –1-1-onto→wf1o 4780   ‘cfv 4781   Isom wiso 4782

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-iso 4796

This theorem is referenced by: (None)