Theorem bnj927 34745

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 3492	. . . . . 6 ⊢ 𝑛 ∈ V
3		bnj927.2	. . . . . 6 ⊢ 𝐶 ∈ V
4	2, 3	fnsn 6636	. . . . 5 ⊢ {⟨𝑛, 𝐶⟩} Fn {𝑛}
5	4	a1i 11	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → {⟨𝑛, 𝐶⟩} Fn {𝑛})
6		disjcsn 9673	. . . . 5 ⊢ (𝑛 ∩ {𝑛}) = ∅
7	6	a1i 11	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅)
8	1, 5, 7	fnund 6694	. . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
9		bnj927.1	. . . 4 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
10	9	fneq1i 6676	. . 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 6401	. . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
13	12	eqeq2i 2753	. . . . 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 6673	. 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 1537   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ∩ cin 3975  ∅c0 4352  {csn 4648  ⟨cop 4654  suc csuc 6397   Fn wfn 6568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-reg 9661

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-suc 6401  df-fun 6575  df-fn 6576

This theorem is referenced by:  bnj941  34748  bnj929  34912