Theorem fnunsn 5361

Description: Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
fnunop.x	⊢ (𝜑 → 𝑋 ∈ V)
fnunop.y	⊢ (𝜑 → 𝑌 ∈ V)
fnunop.f	⊢ (𝜑 → 𝐹 Fn 𝐷)
fnunop.g	⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
fnunop.e	⊢ 𝐸 = (𝐷 ∪ {𝑋})
fnunop.d	⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷)

Assertion

Ref	Expression
fnunsn	⊢ (𝜑 → 𝐺 Fn 𝐸)

Proof of Theorem fnunsn

Step	Hyp	Ref	Expression
1		fnunop.f	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐷)
2		fnunop.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
3		fnunop.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ V)
4		fnsng 5301	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
5	2, 3, 4	syl2anc 411	. . 3 ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6		fnunop.d	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷)
7		disjsn 3680	. . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷)
8	6, 7	sylibr 134	. . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅)
9		fnun 5360	. . 3 ⊢ (((𝐹 Fn 𝐷 ∧ {⟨𝑋, 𝑌⟩} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
10	1, 5, 8, 9	syl21anc 1248	. 2 ⊢ (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
11		fnunop.g	. . . 4 ⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
12	11	fneq1i 5348	. . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
13		fnunop.e	. . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋})
14	13	fneq2i 5349	. . 3 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
15	12, 14	bitri 184	. 2 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
16	10, 15	sylibr 134	1 ⊢ (𝜑 → 𝐺 Fn 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∪ cun 3151   ∩ cin 3152  ∅c0 3446  {csn 3618  ⟨cop 3621   Fn wfn 5249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-fun 5256  df-fn 5257

This theorem is referenced by:  tfrlemisucfn  6377  tfr1onlemsucfn  6393