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Theorem fnunsn 5200
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 5140 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
52, 3, 4syl2anc 408 . . 3 (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6 fnunop.d . . . 4 (𝜑 → ¬ 𝑋𝐷)
7 disjsn 3555 . . . 4 ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐷)
86, 7sylibr 133 . . 3 (𝜑 → (𝐷 ∩ {𝑋}) = ∅)
9 fnun 5199 . . 3 (((𝐹 Fn 𝐷 ∧ {⟨𝑋, 𝑌⟩} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
101, 5, 8, 9syl21anc 1200 . 2 (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
11 fnunop.g . . . 4 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
1211fneq1i 5187 . . 3 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
13 fnunop.e . . . 4 𝐸 = (𝐷 ∪ {𝑋})
1413fneq2i 5188 . . 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 1316  wcel 1465  Vcvv 2660  cun 3039  cin 3040  c0 3333  {csn 3497  cop 3500   Fn wfn 5088
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-fun 5095  df-fn 5096
This theorem is referenced by:  tfrlemisucfn  6189  tfr1onlemsucfn  6205
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