Theorem bnj927 34752

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 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)

Proof of Theorem bnj927

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑓 Fn 𝑛)
2		vex 3440	. . . . . 6 ⊢ 𝑛 ∈ V
3		bnj927.2	. . . . . 6 ⊢ 𝐶 ∈ V
4	2, 3	fnsn 6540	. . . . 5 ⊢ {⟨𝑛, 𝐶⟩} Fn {𝑛}
5	4	a1i 11	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → {⟨𝑛, 𝐶⟩} Fn {𝑛})
6		disjcsn 9499	. . . . 5 ⊢ (𝑛 ∩ {𝑛}) = ∅
7	6	a1i 11	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅)
8	1, 5, 7	fnund 6597	. . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
9		bnj927.1	. . . 4 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
10	9	fneq1i 6579	. . 3 ⊢ (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
11	8, 10	sylibr 234	. 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛}))
12		df-suc 6313	. . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
13	12	eqeq2i 2742	. . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛}))
14	13	biimpi 216	. . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 = (𝑛 ∪ {𝑛}))
15	14	adantr 480	. . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛}))
16	15	fneq2d 6576	. 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝐺 Fn 𝑝 ↔ 𝐺 Fn (𝑛 ∪ {𝑛})))
17	11, 16	mpbird 257	1 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ∪ cun 3901   ∩ cin 3902  ∅c0 4284  {csn 4577  ⟨cop 4583  suc csuc 6309   Fn wfn 6477

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-reg 9484

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-suc 6313  df-fun 6484  df-fn 6485

This theorem is referenced by:  bnj941  34755  bnj929  34919