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Theorem fnunsn 5305
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 (𝜑𝑋 ∈ V)
fnunop.y (𝜑𝑌 ∈ V)
fnunop.f (𝜑𝐹 Fn 𝐷)
fnunop.g 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
fnunop.e 𝐸 = (𝐷 ∪ {𝑋})
fnunop.d (𝜑 → ¬ 𝑋𝐷)
Assertion
Ref Expression
fnunsn (𝜑𝐺 Fn 𝐸)

Proof of Theorem fnunsn
StepHypRef Expression
1 fnunop.f . . 3 (𝜑𝐹 Fn 𝐷)
2 fnunop.x . . . 4 (𝜑𝑋 ∈ V)
3 fnunop.y . . . 4 (𝜑𝑌 ∈ V)
4 fnsng 5245 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
52, 3, 4syl2anc 409 . . 3 (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6 fnunop.d . . . 4 (𝜑 → ¬ 𝑋𝐷)
7 disjsn 3645 . . . 4 ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐷)
86, 7sylibr 133 . . 3 (𝜑 → (𝐷 ∩ {𝑋}) = ∅)
9 fnun 5304 . . 3 (((𝐹 Fn 𝐷 ∧ {⟨𝑋, 𝑌⟩} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
101, 5, 8, 9syl21anc 1232 . 2 (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
11 fnunop.g . . . 4 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
1211fneq1i 5292 . . 3 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
13 fnunop.e . . . 4 𝐸 = (𝐷 ∪ {𝑋})
1413fneq2i 5293 . . 3 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
1512, 14bitri 183 . 2 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
1610, 15sylibr 133 1 (𝜑𝐺 Fn 𝐸)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1348  wcel 2141  Vcvv 2730  cun 3119  cin 3120  c0 3414  {csn 3583  cop 3586   Fn wfn 5193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-fun 5200  df-fn 5201
This theorem is referenced by:  tfrlemisucfn  6303  tfr1onlemsucfn  6319
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