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Theorem fnunop 6658
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 6593 . . . 4 ((𝑋𝑉𝑌𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
52, 3, 4syl2anc 583 . . 3 (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6 fnunop.d . . . 4 (𝜑 → ¬ 𝑋𝐷)
7 disjsn 4710 . . . 4 ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐷)
86, 7sylibr 233 . . 3 (𝜑 → (𝐷 ∩ {𝑋}) = ∅)
91, 5, 8fnund 6657 . 2 (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
10 fnunop.g . . . 4 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
1110fneq1i 6639 . . 3 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
12 fnunop.e . . . 4 𝐸 = (𝐷 ∪ {𝑋})
1312fneq2i 6640 . . 3 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
1411, 13bitri 275 . 2 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
159, 14sylibr 233 1 (𝜑𝐺 Fn 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  cun 3941  cin 3942  c0 4317  {csn 4623  cop 4629   Fn wfn 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-fun 6538  df-fn 6539
This theorem is referenced by:  fineqvac  34626
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