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Theorem bnj927 32148
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 488 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝑓 Fn 𝑛)
2 vex 3447 . . . . . 6 𝑛 ∈ V
3 bnj927.2 . . . . . 6 𝐶 ∈ V
42, 3fnsn 6386 . . . . 5 {⟨𝑛, 𝐶⟩} Fn {𝑛}
54a1i 11 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → {⟨𝑛, 𝐶⟩} Fn {𝑛})
6 bnj521 32115 . . . . 5 (𝑛 ∩ {𝑛}) = ∅
76a1i 11 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅)
81, 5, 7fnund 6439 . . 3 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
9 bnj927.1 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
109fneq1i 6424 . . 3 (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
118, 10sylibr 237 . 2 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛}))
12 df-suc 6169 . . . . . 6 suc 𝑛 = (𝑛 ∪ {𝑛})
1312eqeq2i 2814 . . . . 5 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
1413biimpi 219 . . . 4 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
1514adantr 484 . . 3 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛}))
1615fneq2d 6421 . 2 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝐺 Fn 𝑝𝐺 Fn (𝑛 ∪ {𝑛})))
1711, 16mpbird 260 1 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  cun 3882  cin 3883  c0 4246  {csn 4528  cop 4534  suc csuc 6165   Fn wfn 6323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-reg 9044
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-suc 6169  df-fun 6330  df-fn 6331
This theorem is referenced by:  bnj941  32152  bnj929  32316
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