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Theorem bnj958 32216
Description: Technical lemma for bnj69 32286. 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 2973 . . . . . 6 𝑦𝑓
3 nfcv 2973 . . . . . . . 8 𝑦𝑛
4 bnj958.1 . . . . . . . . 9 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
5 nfiu1 4925 . . . . . . . . 9 𝑦 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
64, 5nfcxfr 2971 . . . . . . . 8 𝑦𝐶
73, 6nfop 4791 . . . . . . 7 𝑦𝑛, 𝐶
87nfsn 4615 . . . . . 6 𝑦{⟨𝑛, 𝐶⟩}
92, 8nfun 4116 . . . . 5 𝑦(𝑓 ∪ {⟨𝑛, 𝐶⟩})
101, 9nfcxfr 2971 . . . 4 𝑦𝐺
11 nfcv 2973 . . . 4 𝑦𝑖
1210, 11nffv 6652 . . 3 𝑦(𝐺𝑖)
1312nfeq1 2988 . 2 𝑦(𝐺𝑖) = (𝑓𝑖)
1413nf5ri 2195 1 ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535   = wceq 1537  cun 3907  {csn 4539  cop 4545   ciun 4891  cfv 6327   predc-bnj14 31962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-iota 6286  df-fv 6335
This theorem is referenced by:  bnj966  32220  bnj967  32221
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