Theorem bnj941 33721

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 4873	. . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ⟨𝑛, 𝐶⟩ = ⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩)
3	2	sneqd 4639	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {⟨𝑛, 𝐶⟩} = {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
4	3	uneq2d 4162	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
5	1, 4	eqtrid 2785	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
6	5	fneq1d 6639	. . 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 2733	. . 3 ⊢ (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
9		0ex 5306	. . . 4 ⊢ ∅ ∈ V
10	9	elimel 4596	. . 3 ⊢ if(𝐶 ∈ V, 𝐶, ∅) ∈ V
11	8, 10	bnj927 33718	. 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) Fn 𝑝)
12	7, 11	dedth 4585	1 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∪ cun 3945  ∅c0 4321  ifcif 4527  {csn 4627  ⟨cop 4633  suc csuc 6363   Fn wfn 6535

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-reg 9583

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-suc 6367  df-fun 6542  df-fn 6543

This theorem is referenced by:  bnj945  33722  bnj910  33897