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Theorem fnressn 6330
 Description: A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
fnressn ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})

Proof of Theorem fnressn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 4135 . . . . . 6 (𝑥 = 𝐵 → {𝑥} = {𝐵})
21reseq2d 5317 . . . . 5 (𝑥 = 𝐵 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3 fveq2 6103 . . . . . . 7 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
4 opeq12 4342 . . . . . . 7 ((𝑥 = 𝐵 ∧ (𝐹𝑥) = (𝐹𝐵)) → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
53, 4mpdan 699 . . . . . 6 (𝑥 = 𝐵 → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
65sneqd 4137 . . . . 5 (𝑥 = 𝐵 → {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝐵, (𝐹𝐵)⟩})
72, 6eqeq12d 2625 . . . 4 (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩} ↔ (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩}))
87imbi2d 329 . . 3 (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}) ↔ (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})))
9 vex 3176 . . . . . . 7 𝑥 ∈ V
109snss 4259 . . . . . 6 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
11 fnssres 5918 . . . . . 6 ((𝐹 Fn 𝐴 ∧ {𝑥} ⊆ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
1210, 11sylan2b 491 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
13 dffn2 5960 . . . . . 6 ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}):{𝑥}⟶V)
149fsn2 6309 . . . . . 6 ((𝐹 ↾ {𝑥}):{𝑥}⟶V ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
15 fvex 6113 . . . . . . . 8 ((𝐹 ↾ {𝑥})‘𝑥) ∈ V
1615biantrur 526 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
17 vsnid 4156 . . . . . . . . . . 11 𝑥 ∈ {𝑥}
18 fvres 6117 . . . . . . . . . . 11 (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥))
1917, 18ax-mp 5 . . . . . . . . . 10 ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥)
2019opeq2i 4344 . . . . . . . . 9 𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩
2120sneqi 4136 . . . . . . . 8 {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹𝑥)⟩}
2221eqeq2i 2622 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2316, 22bitr3i 265 . . . . . 6 ((((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}) ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2413, 14, 233bitri 285 . . . . 5 ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2512, 24sylib 207 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2625expcom 450 . . 3 (𝑥𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}))
278, 26vtoclga 3245 . 2 (𝐵𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩}))
2827impcom 445 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  {csn 4125  ⟨cop 4131   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  funressn  6331  fressnfv  6332  fnsnsplit  6355  canthp1lem2  9354  fseq1p1m1  12283  dprd2da  18264  dmdprdpr  18271  dprdpr  18272  dpjlem  18273  pgpfaclem1  18303  islindf4  19996  xpstopnlem1  21422  ptcmpfi  21426  2pthlem1  26125  eupath2lem3  26506  subfacp1lem5  30420  cvmliftlem10  30530  poimirlem9  32588  resunimafz0  40368
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