Theorem bnj927 34781

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 6539	. . . . 5 ⊢ {⟨𝑛, 𝐶⟩} Fn {𝑛}
5	4	a1i 11	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → {⟨𝑛, 𝐶⟩} Fn {𝑛})
6		disjcsn 9493	. . . . 5 ⊢ (𝑛 ∩ {𝑛}) = ∅
7	6	a1i 11	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅)
8	1, 5, 7	fnund 6596	. . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
9		bnj927.1	. . . 4 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
10	9	fneq1i 6578	. . 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 6312	. . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
13	12	eqeq2i 2744	. . . . 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 6575	. 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 1541   ∈ wcel 2111  Vcvv 3436   ∪ cun 3895   ∩ cin 3896  ∅c0 4280  {csn 4573  ⟨cop 4579  suc csuc 6308   Fn wfn 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-reg 9478

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-suc 6312  df-fun 6483  df-fn 6484

This theorem is referenced by:  bnj941  34784  bnj929  34948