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Theorem bnj927 30965
 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 476 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝑓 Fn 𝑛)
2 vex 3234 . . . . . 6 𝑛 ∈ V
3 bnj927.2 . . . . . 6 𝐶 ∈ V
42, 3fnsn 5984 . . . . 5 {⟨𝑛, 𝐶⟩} Fn {𝑛}
54a1i 11 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → {⟨𝑛, 𝐶⟩} Fn {𝑛})
6 bnj521 30931 . . . . 5 (𝑛 ∩ {𝑛}) = ∅
76a1i 11 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅)
8 fnun 6035 . . . 4 (((𝑓 Fn 𝑛 ∧ {⟨𝑛, 𝐶⟩} Fn {𝑛}) ∧ (𝑛 ∩ {𝑛}) = ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
91, 5, 7, 8syl21anc 1365 . . 3 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
10 bnj927.1 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
1110fneq1i 6023 . . 3 (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
129, 11sylibr 224 . 2 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛}))
13 df-suc 5767 . . . . . 6 suc 𝑛 = (𝑛 ∪ {𝑛})
1413eqeq2i 2663 . . . . 5 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
1514biimpi 206 . . . 4 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
1615adantr 480 . . 3 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛}))
1716fneq2d 6020 . 2 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝐺 Fn 𝑝𝐺 Fn (𝑛 ∪ {𝑛})))
1812, 17mpbird 247 1 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605   ∩ cin 3606  ∅c0 3948  {csn 4210  ⟨cop 4216  suc csuc 5763   Fn wfn 5921 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-suc 5767  df-fun 5928  df-fn 5929 This theorem is referenced by:  bnj941  30969  bnj929  31132
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