Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj958 Structured version   Visualization version   GIF version

Theorem bnj958 31479
Description: Technical lemma for bnj69 31547. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj958.1 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj958.2 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
Assertion
Ref Expression
bnj958 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
Distinct variable groups:   𝑦,𝑓   𝑦,𝑖   𝑦,𝑛
Allowed substitution hints:   𝐴(𝑦,𝑓,𝑖,𝑚,𝑛)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛)   𝑅(𝑦,𝑓,𝑖,𝑚,𝑛)   𝐺(𝑦,𝑓,𝑖,𝑚,𝑛)

Proof of Theorem bnj958
StepHypRef Expression
1 bnj958.2 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
2 nfcv 2907 . . . . . 6 𝑦𝑓
3 nfcv 2907 . . . . . . . 8 𝑦𝑛
4 bnj958.1 . . . . . . . . 9 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
5 nfiu1 4708 . . . . . . . . 9 𝑦 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
64, 5nfcxfr 2905 . . . . . . . 8 𝑦𝐶
73, 6nfop 4577 . . . . . . 7 𝑦𝑛, 𝐶
87nfsn 4400 . . . . . 6 𝑦{⟨𝑛, 𝐶⟩}
92, 8nfun 3933 . . . . 5 𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
101, 9nfcxfr 2905 . . . 4 𝑦𝐺
11 nfcv 2907 . . . 4 𝑦𝑖
1210, 11nffv 6389 . . 3 𝑦(𝐺𝑖)
1312nfeq1 2921 . 2 𝑦(𝐺𝑖) = (𝑓𝑖)
1413nf5ri 2227 1 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650   = wceq 1652  cun 3732  {csn 4336  cop 4342   ciun 4678  cfv 6070   predc-bnj14 31226
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-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-iota 6033  df-fv 6078
This theorem is referenced by:  bnj966  31483  bnj967  31484
  Copyright terms: Public domain W3C validator