Theorem bnj927 32749

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 485	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑓 Fn 𝑛)
2		vex 3436	. . . . . 6 ⊢ 𝑛 ∈ V
3		bnj927.2	. . . . . 6 ⊢ 𝐶 ∈ V
4	2, 3	fnsn 6492	. . . . 5 ⊢ {⟨𝑛, 𝐶⟩} Fn {𝑛}
5	4	a1i 11	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → {⟨𝑛, 𝐶⟩} Fn {𝑛})
6		bnj521 32716	. . . . 5 ⊢ (𝑛 ∩ {𝑛}) = ∅
7	6	a1i 11	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅)
8	1, 5, 7	fnund 6546	. . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
9		bnj927.1	. . . 4 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
10	9	fneq1i 6530	. . 3 ⊢ (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
11	8, 10	sylibr 233	. 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛}))
12		df-suc 6272	. . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
13	12	eqeq2i 2751	. . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛}))
14	13	biimpi 215	. . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 = (𝑛 ∪ {𝑛}))
15	14	adantr 481	. . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛}))
16	15	fneq2d 6527	. 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝐺 Fn 𝑝 ↔ 𝐺 Fn (𝑛 ∪ {𝑛})))
17	11, 16	mpbird 256	1 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885   ∩ cin 3886  ∅c0 4256  {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:  bnj941  32752  bnj929  32916