Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfof GIF version

Theorem nfof 5996
 Description: Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypothesis
Ref Expression
nfof.1 𝑥𝑅
Assertion
Ref Expression
nfof 𝑥𝑓 𝑅

Proof of Theorem nfof
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-of 5991 . 2 𝑓 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢𝑤)𝑅(𝑣𝑤))))
2 nfcv 2282 . . 3 𝑥V
3 nfcv 2282 . . . 4 𝑥(dom 𝑢 ∩ dom 𝑣)
4 nfcv 2282 . . . . 5 𝑥(𝑢𝑤)
5 nfof.1 . . . . 5 𝑥𝑅
6 nfcv 2282 . . . . 5 𝑥(𝑣𝑤)
74, 5, 6nfov 5810 . . . 4 𝑥((𝑢𝑤)𝑅(𝑣𝑤))
83, 7nfmpt 4029 . . 3 𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢𝑤)𝑅(𝑣𝑤)))
92, 2, 8nfmpo 5849 . 2 𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢𝑤)𝑅(𝑣𝑤))))
101, 9nfcxfr 2279 1 𝑥𝑓 𝑅
 Colors of variables: wff set class Syntax hints:  Ⅎwnfc 2269  Vcvv 2690   ∩ cin 3076   ↦ cmpt 3998  dom cdm 4548  ‘cfv 5132  (class class class)co 5783   ∈ cmpo 5785   ∘𝑓 cof 5989 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2692  df-un 3081  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-mpt 4000  df-iota 5097  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-of 5991 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator