New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  nfiso GIF version

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 xH
nfiso.2 xR
nfiso.3 xS
nfiso.4 xA
nfiso.5 xB
Assertion
Ref Expression
nfiso x H Isom R, S (A, B)

Proof of Theorem nfiso
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iso 4796 . 2 (H Isom R, S (A, B) ↔ (H:A1-1-ontoB y A z A (yRz ↔ (Hy)S(Hz))))
2 nfiso.1 . . . 4 xH
3 nfiso.4 . . . 4 xA
4 nfiso.5 . . . 4 xB
52, 3, 4nff1o 5285 . . 3 x H:A1-1-ontoB
6 nfcv 2489 . . . . . . 7 xy
7 nfiso.2 . . . . . . 7 xR
8 nfcv 2489 . . . . . . 7 xz
96, 7, 8nfbr 4683 . . . . . 6 x yRz
102, 6nffv 5334 . . . . . . 7 x(Hy)
11 nfiso.3 . . . . . . 7 xS
122, 8nffv 5334 . . . . . . 7 x(Hz)
1310, 11, 12nfbr 4683 . . . . . 6 x(Hy)S(Hz)
149, 13nfbi 1834 . . . . 5 x(yRz ↔ (Hy)S(Hz))
153, 14nfral 2667 . . . 4 xz A (yRz ↔ (Hy)S(Hz))
163, 15nfral 2667 . . 3 xy A z A (yRz ↔ (Hy)S(Hz))
175, 16nfan 1824 . 2 x(H:A1-1-ontoB y A z A (yRz ↔ (Hy)S(Hz)))
181, 17nfxfr 1570 1 x H Isom R, S (A, B)
 Colors of variables: wff setvar class 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)
 Copyright terms: Public domain W3C validator