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Theorem fnressn 6913
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 4569 . . . . . 6 (𝑥 = 𝐵 → {𝑥} = {𝐵})
21reseq2d 5846 . . . . 5 (𝑥 = 𝐵 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3 fveq2 6663 . . . . . . 7 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
4 opeq12 4797 . . . . . . 7 ((𝑥 = 𝐵 ∧ (𝐹𝑥) = (𝐹𝐵)) → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
53, 4mpdan 685 . . . . . 6 (𝑥 = 𝐵 → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
65sneqd 4571 . . . . 5 (𝑥 = 𝐵 → {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝐵, (𝐹𝐵)⟩})
72, 6eqeq12d 2835 . . . 4 (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩} ↔ (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩}))
87imbi2d 343 . . 3 (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}) ↔ (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})))
9 vex 3496 . . . . . . 7 𝑥 ∈ V
109snss 4710 . . . . . 6 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
11 fnssres 6463 . . . . . 6 ((𝐹 Fn 𝐴 ∧ {𝑥} ⊆ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
1210, 11sylan2b 595 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
13 dffn2 6509 . . . . . 6 ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}):{𝑥}⟶V)
149fsn2 6891 . . . . . 6 ((𝐹 ↾ {𝑥}):{𝑥}⟶V ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
15 fvex 6676 . . . . . . . 8 ((𝐹 ↾ {𝑥})‘𝑥) ∈ V
1615biantrur 533 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
17 vsnid 4594 . . . . . . . . . . 11 𝑥 ∈ {𝑥}
18 fvres 6682 . . . . . . . . . . 11 (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥))
1917, 18ax-mp 5 . . . . . . . . . 10 ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥)
2019opeq2i 4799 . . . . . . . . 9 𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩
2120sneqi 4570 . . . . . . . 8 {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹𝑥)⟩}
2221eqeq2i 2832 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2316, 22bitr3i 279 . . . . . 6 ((((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}) ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2413, 14, 233bitri 299 . . . . 5 ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2512, 24sylib 220 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2625expcom 416 . . 3 (𝑥𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}))
278, 26vtoclga 3572 . 2 (𝐵𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩}))
2827impcom 410 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  Vcvv 3493  wss 3934  {csn 4559  cop 4565  cres 5550   Fn wfn 6343  wf 6344  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  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-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  funressn  6914  fressnfv  6915  fnsnsplit  6939  canthp1lem2  10067  fseq1p1m1  12973  resunimafz0  13795  dprd2da  19156  dmdprdpr  19163  dprdpr  19164  dpjlem  19165  pgpfaclem1  19195  islindf4  20974  xpstopnlem1  22409  ptcmpfi  22413  subfacp1lem5  32419  cvmliftlem10  32529  nosupbnd2lem1  33203  poimirlem9  34888
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