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Theorem bnj927 34783
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 3484 . . . . . 6 𝑛 ∈ V
3 bnj927.2 . . . . . 6 𝐶 ∈ V
42, 3fnsn 6624 . . . . 5 {⟨𝑛, 𝐶⟩} Fn {𝑛}
54a1i 11 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → {⟨𝑛, 𝐶⟩} Fn {𝑛})
6 disjcsn 9644 . . . . 5 (𝑛 ∩ {𝑛}) = ∅
76a1i 11 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅)
81, 5, 7fnund 6683 . . 3 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
9 bnj927.1 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
109fneq1i 6665 . . 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 6390 . . . . . 6 suc 𝑛 = (𝑛 ∪ {𝑛})
1312eqeq2i 2750 . . . . 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 6662 . 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 1540  wcel 2108  Vcvv 3480  cun 3949  cin 3950  c0 4333  {csn 4626  cop 4632  suc csuc 6386   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-suc 6390  df-fun 6563  df-fn 6564
This theorem is referenced by:  bnj941  34786  bnj929  34950
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