Theorem bnj927 31649

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 477	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑓 Fn 𝑛)
2		vex 3412	. . . . . 6 ⊢ 𝑛 ∈ V
3		bnj927.2	. . . . . 6 ⊢ 𝐶 ∈ V
4	2, 3	fnsn 6239	. . . . 5 ⊢ {⟨𝑛, 𝐶⟩} Fn {𝑛}
5	4	a1i 11	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → {⟨𝑛, 𝐶⟩} Fn {𝑛})
6		bnj521 31616	. . . . 5 ⊢ (𝑛 ∩ {𝑛}) = ∅
7	6	a1i 11	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅)
8		fnun 6290	. . . 4 ⊢ (((𝑓 Fn 𝑛 ∧ {⟨𝑛, 𝐶⟩} Fn {𝑛}) ∧ (𝑛 ∩ {𝑛}) = ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
9	1, 5, 7, 8	syl21anc 825	. . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
10		bnj927.1	. . . 4 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
11	10	fneq1i 6277	. . 3 ⊢ (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
12	9, 11	sylibr 226	. 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛}))
13		df-suc 6029	. . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
14	13	eqeq2i 2784	. . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛}))
15	14	biimpi 208	. . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 = (𝑛 ∪ {𝑛}))
16	15	adantr 473	. . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛}))
17	16	fneq2d 6274	. 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝐺 Fn 𝑝 ↔ 𝐺 Fn (𝑛 ∪ {𝑛})))
18	12, 17	mpbird 249	1 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2048  Vcvv 3409   ∪ cun 3823   ∩ cin 3824  ∅c0 4173  {csn 4435  ⟨cop 4441  suc csuc 6025   Fn wfn 6177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180  ax-reg 8843

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-suc 6029  df-fun 6184  df-fn 6185

This theorem is referenced by:  bnj941  31653  bnj929  31816