MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnunsn Structured version   Visualization version   GIF version

Theorem fnunsn 6457
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 6399 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
52, 3, 4syl2anc 584 . . 3 (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6 fnunop.d . . . 4 (𝜑 → ¬ 𝑋𝐷)
7 disjsn 4639 . . . 4 ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐷)
86, 7sylibr 235 . . 3 (𝜑 → (𝐷 ∩ {𝑋}) = ∅)
9 fnun 6456 . . 3 (((𝐹 Fn 𝐷 ∧ {⟨𝑋, 𝑌⟩} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
101, 5, 8, 9syl21anc 833 . 2 (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
11 fnunop.g . . . 4 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
1211fneq1i 6443 . . 3 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
13 fnunop.e . . . 4 𝐸 = (𝐷 ∪ {𝑋})
1413fneq2i 6444 . . 3 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
1512, 14bitri 276 . 2 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
1610, 15sylibr 235 1 (𝜑𝐺 Fn 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  cun 3931  cin 3932  c0 4288  {csn 4557  cop 4563   Fn wfn 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-fun 6350  df-fn 6351
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator