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Theorem fnunsn 5190
 Description: Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.)
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
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnunop.f . . 3 (φF Fn D)
2 fnunop.x . . . 4 (φX V)
3 fnunop.y . . . 4 (φY V)
4 opeq1 4578 . . . . . . 7 (x = Xx, y = X, y)
54sneqd 3746 . . . . . 6 (x = X → {x, y} = {X, y})
6 sneq 3744 . . . . . 6 (x = X → {x} = {X})
75, 6fneq12d 5177 . . . . 5 (x = X → ({x, y} Fn {x} ↔ {X, y} Fn {X}))
8 opeq2 4579 . . . . . . 7 (y = YX, y = X, Y)
98sneqd 3746 . . . . . 6 (y = Y → {X, y} = {X, Y})
109fneq1d 5175 . . . . 5 (y = Y → ({X, y} Fn {X} ↔ {X, Y} Fn {X}))
11 vex 2862 . . . . . 6 x V
12 vex 2862 . . . . . 6 y V
1311, 12fnsn 5152 . . . . 5 {x, y} Fn {x}
147, 10, 13vtocl2g 2918 . . . 4 ((X V Y V) → {X, Y} Fn {X})
152, 3, 14syl2anc 642 . . 3 (φ → {X, Y} Fn {X})
16 fnunop.d . . . 4 (φ → ¬ X D)
17 disjsn 3786 . . . 4 ((D ∩ {X}) = ↔ ¬ X D)
1816, 17sylibr 203 . . 3 (φ → (D ∩ {X}) = )
19 fnun 5189 . . 3 (((F Fn D {X, Y} Fn {X}) (D ∩ {X}) = ) → (F ∪ {X, Y}) Fn (D ∪ {X}))
201, 15, 18, 19syl21anc 1181 . 2 (φ → (F ∪ {X, Y}) Fn (D ∪ {X}))
21 fnunop.g . . . 4 G = (F ∪ {X, Y})
2221fneq1i 5178 . . 3 (G Fn E ↔ (F ∪ {X, Y}) Fn E)
23 fnunop.e . . . 4 E = (D ∪ {X})
2423fneq2i 5179 . . 3 ((F ∪ {X, Y}) Fn E ↔ (F ∪ {X, Y}) Fn (D ∪ {X}))
2522, 24bitri 240 . 2 (G Fn E ↔ (F ∪ {X, Y}) Fn (D ∪ {X}))
2620, 25sylibr 203 1 (φG Fn E)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  ⟨cop 4561   Fn wfn 4776 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790 This theorem is referenced by: (None)
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