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 31136
Description: Technical lemma for bnj69 31204. 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 2793 . . . . . 6 𝑦𝑓
3 nfcv 2793 . . . . . . . 8 𝑦𝑛
4 bnj958.1 . . . . . . . . 9 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
5 nfiu1 4582 . . . . . . . . 9 𝑦 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
64, 5nfcxfr 2791 . . . . . . . 8 𝑦𝐶
73, 6nfop 4449 . . . . . . 7 𝑦𝑛, 𝐶
87nfsn 4274 . . . . . 6 𝑦{⟨𝑛, 𝐶⟩}
92, 8nfun 3802 . . . . 5 𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
101, 9nfcxfr 2791 . . . 4 𝑦𝐺
11 nfcv 2793 . . . 4 𝑦𝑖
1210, 11nffv 6236 . . 3 𝑦(𝐺𝑖)
1312nfeq1 2807 . 2 𝑦(𝐺𝑖) = (𝑓𝑖)
1413nf5ri 2103 1 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521   = wceq 1523  cun 3605  {csn 4210  cop 4216   ciun 4552  cfv 5926   predc-bnj14 30882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-iota 5889  df-fv 5934
This theorem is referenced by:  bnj966  31140  bnj967  31141
  Copyright terms: Public domain W3C validator