Theorem fnunop 6670

Description: Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 16-Aug-2024.)

Hypotheses

Ref	Expression
fnunop.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
fnunop.y	⊢ (𝜑 → 𝑌 ∈ 𝑊)
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
fnunop	⊢ (𝜑 → 𝐺 Fn 𝐸)

Proof of Theorem fnunop

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 ∈ 𝑊)
4		fnsng 6605	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
5	2, 3, 4	syl2anc 583	. . 3 ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6		fnunop.d	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷)
7		disjsn 4716	. . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷)
8	6, 7	sylibr 233	. . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅)
9	1, 5, 8	fnund 6669	. 2 ⊢ (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
10		fnunop.g	. . . 4 ⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
11	10	fneq1i 6651	. . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
12		fnunop.e	. . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋})
13	12	fneq2i 6652	. . 3 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
14	11, 13	bitri 275	. 2 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
15	9, 14	sylibr 233	1 ⊢ (𝜑 → 𝐺 Fn 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534   ∈ wcel 2099   ∪ cun 3945   ∩ cin 3946  ∅c0 4323  {csn 4629  ⟨cop 4635   Fn wfn 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-fun 6550  df-fn 6551

This theorem is referenced by:  fineqvac  34717