Theorem bnj941 32752

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj941.1	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})

Assertion

Ref	Expression
bnj941	⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))

Proof of Theorem bnj941

Step	Hyp	Ref	Expression
1		bnj941.1	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
2		opeq2 4805	. . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ⟨𝑛, 𝐶⟩ = ⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩)
3	2	sneqd 4573	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {⟨𝑛, 𝐶⟩} = {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
4	3	uneq2d 4097	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
5	1, 4	eqtrid 2790	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
6	5	fneq1d 6526	. . 3 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝐺 Fn 𝑝 ↔ (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) Fn 𝑝))
7	6	imbi2d 341	. 2 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) Fn 𝑝)))
8		eqid 2738	. . 3 ⊢ (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
9		0ex 5231	. . . 4 ⊢ ∅ ∈ V
10	9	elimel 4528	. . 3 ⊢ if(𝐶 ∈ V, 𝐶, ∅) ∈ V
11	8, 10	bnj927 32749	. 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) Fn 𝑝)
12	7, 11	dedth 4517	1 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885  ∅c0 4256  ifcif 4459  {csn 4561  ⟨cop 4567  suc csuc 6268   Fn wfn 6428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-reg 9351

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-suc 6272  df-fun 6435  df-fn 6436

This theorem is referenced by:  bnj945  32753  bnj910  32928