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Theorem bnj958 30771
Description: Technical lemma for bnj69 30839. 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 2761 . . . . . 6 𝑦𝑓
3 nfcv 2761 . . . . . . . 8 𝑦𝑛
4 bnj958.1 . . . . . . . . 9 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
5 nfiu1 4523 . . . . . . . . 9 𝑦 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
64, 5nfcxfr 2759 . . . . . . . 8 𝑦𝐶
73, 6nfop 4393 . . . . . . 7 𝑦𝑛, 𝐶
87nfsn 4220 . . . . . 6 𝑦{⟨𝑛, 𝐶⟩}
92, 8nfun 3753 . . . . 5 𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
101, 9nfcxfr 2759 . . . 4 𝑦𝐺
11 nfcv 2761 . . . 4 𝑦𝑖
1210, 11nffv 6165 . . 3 𝑦(𝐺𝑖)
1312nfeq1 2774 . 2 𝑦(𝐺𝑖) = (𝑓𝑖)
1413nf5ri 2063 1 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478   = wceq 1480  cun 3558  {csn 4155  cop 4161   ciun 4492  cfv 5857   predc-bnj14 30514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-iota 5820  df-fv 5865
This theorem is referenced by:  bnj966  30775  bnj967  30776
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