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Theorem fnressn 6568
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 4326 . . . . . 6 (𝑥 = 𝐵 → {𝑥} = {𝐵})
21reseq2d 5534 . . . . 5 (𝑥 = 𝐵 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3 fveq2 6332 . . . . . . 7 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
4 opeq12 4541 . . . . . . 7 ((𝑥 = 𝐵 ∧ (𝐹𝑥) = (𝐹𝐵)) → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
53, 4mpdan 667 . . . . . 6 (𝑥 = 𝐵 → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
65sneqd 4328 . . . . 5 (𝑥 = 𝐵 → {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝐵, (𝐹𝐵)⟩})
72, 6eqeq12d 2786 . . . 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 3354 . . . . . . 7 𝑥 ∈ V
109snss 4451 . . . . . 6 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
11 fnssres 6144 . . . . . 6 ((𝐹 Fn 𝐴 ∧ {𝑥} ⊆ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
1210, 11sylan2b 581 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
13 dffn2 6187 . . . . . 6 ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}):{𝑥}⟶V)
149fsn2 6546 . . . . . 6 ((𝐹 ↾ {𝑥}):{𝑥}⟶V ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
15 fvex 6342 . . . . . . . 8 ((𝐹 ↾ {𝑥})‘𝑥) ∈ V
1615biantrur 520 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
17 vsnid 4348 . . . . . . . . . . 11 𝑥 ∈ {𝑥}
18 fvres 6348 . . . . . . . . . . 11 (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥))
1917, 18ax-mp 5 . . . . . . . . . 10 ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥)
2019opeq2i 4543 . . . . . . . . 9 𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩
2120sneqi 4327 . . . . . . . 8 {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹𝑥)⟩}
2221eqeq2i 2783 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2316, 22bitr3i 266 . . . . . 6 ((((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}) ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2413, 14, 233bitri 286 . . . . 5 ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2512, 24sylib 208 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2625expcom 398 . . 3 (𝑥𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}))
278, 26vtoclga 3423 . 2 (𝐵𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩}))
2827impcom 394 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723  {csn 4316  cop 4322  cres 5251   Fn wfn 6026  wf 6027  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  funressn  6569  fressnfv  6570  fnsnsplit  6594  canthp1lem2  9677  fseq1p1m1  12621  resunimafz0  13431  dprd2da  18649  dmdprdpr  18656  dprdpr  18657  dpjlem  18658  pgpfaclem1  18688  islindf4  20394  xpstopnlem1  21833  ptcmpfi  21837  subfacp1lem5  31504  cvmliftlem10  31614  nosupbnd2lem1  32198  poimirlem9  33751
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