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Theorem fnunop 6513
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 (𝜑𝑋𝑉)
fnunop.y (𝜑𝑌𝑊)
fnunop.f (𝜑𝐹 Fn 𝐷)
fnunop.g 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
fnunop.e 𝐸 = (𝐷 ∪ {𝑋})
fnunop.d (𝜑 → ¬ 𝑋𝐷)
Assertion
Ref Expression
fnunop (𝜑𝐺 Fn 𝐸)

Proof of Theorem fnunop
StepHypRef Expression
1 fnunop.f . . 3 (𝜑𝐹 Fn 𝐷)
2 fnunop.x . . . 4 (𝜑𝑋𝑉)
3 fnunop.y . . . 4 (𝜑𝑌𝑊)
4 fnsng 6452 . . . 4 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
52, 3, 4syl2anc 587 . . 3 (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6 fnunop.d . . . 4 (𝜑 → ¬ 𝑋𝐷)
7 disjsn 4643 . . . 4 ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐷)
86, 7sylibr 237 . . 3 (𝜑 → (𝐷 ∩ {𝑋}) = ∅)
91, 5, 8fnund 6512 . 2 (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
10 fnunop.g . . . 4 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
1110fneq1i 6496 . . 3 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
12 fnunop.e . . . 4 𝐸 = (𝐷 ∪ {𝑋})
1312fneq2i 6497 . . 3 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
1411, 13bitri 278 . 2 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
159, 14sylibr 237 1 (𝜑𝐺 Fn 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1543  wcel 2112  cun 3880  cin 3881  c0 4253  {csn 4557  cop 4563   Fn wfn 6395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-br 5070  df-opab 5132  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-fun 6402  df-fn 6403
This theorem is referenced by:  fineqvac  32807
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