MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfofr Structured version   Visualization version   GIF version

Theorem nfofr 7101
Description: Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypothesis
Ref Expression
nfof.1 𝑥𝑅
Assertion
Ref Expression
nfofr 𝑥𝑟 𝑅

Proof of Theorem nfofr
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ofr 7096 . 2 𝑟 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)}
2 nfcv 2907 . . . 4 𝑥(dom 𝑢 ∩ dom 𝑣)
3 nfcv 2907 . . . . 5 𝑥(𝑢𝑤)
4 nfof.1 . . . . 5 𝑥𝑅
5 nfcv 2907 . . . . 5 𝑥(𝑣𝑤)
63, 4, 5nfbr 4856 . . . 4 𝑥(𝑢𝑤)𝑅(𝑣𝑤)
72, 6nfral 3092 . . 3 𝑥𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)
87nfopab 4877 . 2 𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)}
91, 8nfcxfr 2905 1 𝑥𝑟 𝑅
Colors of variables: wff setvar class
Syntax hints:  wnfc 2894  wral 3055  cin 3731   class class class wbr 4809  {copab 4871  dom cdm 5277  cfv 6068  𝑟 cofr 7094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-ofr 7096
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator