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Theorem bnj927 34762
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj927.1 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj927.2 𝐶 ∈ V
Assertion
Ref Expression
bnj927 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)

Proof of Theorem bnj927
StepHypRef Expression
1 simpr 484 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝑓 Fn 𝑛)
2 vex 3482 . . . . . 6 𝑛 ∈ V
3 bnj927.2 . . . . . 6 𝐶 ∈ V
42, 3fnsn 6626 . . . . 5 {⟨𝑛, 𝐶⟩} Fn {𝑛}
54a1i 11 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → {⟨𝑛, 𝐶⟩} Fn {𝑛})
6 disjcsn 9642 . . . . 5 (𝑛 ∩ {𝑛}) = ∅
76a1i 11 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅)
81, 5, 7fnund 6684 . . 3 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
9 bnj927.1 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
109fneq1i 6666 . . 3 (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
118, 10sylibr 234 . 2 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛}))
12 df-suc 6392 . . . . . 6 suc 𝑛 = (𝑛 ∪ {𝑛})
1312eqeq2i 2748 . . . . 5 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
1413biimpi 216 . . . 4 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
1514adantr 480 . . 3 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛}))
1615fneq2d 6663 . 2 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝐺 Fn 𝑝𝐺 Fn (𝑛 ∪ {𝑛})))
1711, 16mpbird 257 1 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  c0 4339  {csn 4631  cop 4637  suc csuc 6388   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-suc 6392  df-fun 6565  df-fn 6566
This theorem is referenced by:  bnj941  34765  bnj929  34929
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