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Theorem fnunsn 4949
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 → ¬ 𝑋 𝐷)
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 4890 . . . 4 ((𝑋 V 𝑌 V) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
52, 3, 4syl2anc 391 . . 3 (φ → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6 fnunop.d . . . 4 (φ → ¬ 𝑋 𝐷)
7 disjsn 3423 . . . 4 ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 𝐷)
86, 7sylibr 137 . . 3 (φ → (𝐷 ∩ {𝑋}) = ∅)
9 fnun 4948 . . 3 (((𝐹 Fn 𝐷 {⟨𝑋, 𝑌⟩} Fn {𝑋}) (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
101, 5, 8, 9syl21anc 1133 . 2 (φ → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
11 fnunop.g . . . 4 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
1211fneq1i 4936 . . 3 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
13 fnunop.e . . . 4 𝐸 = (𝐷 ∪ {𝑋})
1413fneq2i 4937 . . 3 ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
1512, 14bitri 173 . 2 (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
1610, 15sylibr 137 1 (φ𝐺 Fn 𝐸)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1242   wcel 1390  Vcvv 2551  cun 2909  cin 2910  c0 3218  {csn 3367  cop 3370   Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848
This theorem is referenced by:  tfrlemisucfn  5879
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