Theorem fnunop 6634

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 6568	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
5	2, 3, 4	syl2anc 584	. . 3 ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6		fnunop.d	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷)
7		disjsn 4675	. . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷)
8	6, 7	sylibr 234	. . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅)
9	1, 5, 8	fnund 6633	. 2 ⊢ (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
10		fnunop.g	. . . 4 ⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
11	10	fneq1i 6615	. . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
12		fnunop.e	. . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋})
13	12	fneq2i 6616	. . 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 234	1 ⊢ (𝜑 → 𝐺 Fn 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ∪ cun 3912   ∩ cin 3913  ∅c0 4296  {csn 4589  ⟨cop 4595   Fn wfn 6506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-fun 6513  df-fn 6514

This theorem is referenced by:  fineqvac  35087