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Theorem fnressn 6308
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 4134 . . . . . 6 (𝑥 = 𝐵 → {𝑥} = {𝐵})
21reseq2d 5304 . . . . 5 (𝑥 = 𝐵 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3 fveq2 6088 . . . . . . 7 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
4 opeq12 4336 . . . . . . 7 ((𝑥 = 𝐵 ∧ (𝐹𝑥) = (𝐹𝐵)) → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
53, 4mpdan 698 . . . . . 6 (𝑥 = 𝐵 → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
65sneqd 4136 . . . . 5 (𝑥 = 𝐵 → {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝐵, (𝐹𝐵)⟩})
72, 6eqeq12d 2624 . . . 4 (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩} ↔ (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩}))
87imbi2d 328 . . 3 (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}) ↔ (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})))
9 vex 3175 . . . . . . 7 𝑥 ∈ V
109snss 4258 . . . . . 6 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
11 fnssres 5904 . . . . . 6 ((𝐹 Fn 𝐴 ∧ {𝑥} ⊆ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
1210, 11sylan2b 490 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
13 dffn2 5946 . . . . . 6 ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}):{𝑥}⟶V)
149fsn2 6294 . . . . . 6 ((𝐹 ↾ {𝑥}):{𝑥}⟶V ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
15 fvex 6098 . . . . . . . 8 ((𝐹 ↾ {𝑥})‘𝑥) ∈ V
1615biantrur 525 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
17 vsnid 4155 . . . . . . . . . . 11 𝑥 ∈ {𝑥}
18 fvres 6102 . . . . . . . . . . 11 (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥))
1917, 18ax-mp 5 . . . . . . . . . 10 ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥)
2019opeq2i 4338 . . . . . . . . 9 𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩
2120sneqi 4135 . . . . . . . 8 {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹𝑥)⟩}
2221eqeq2i 2621 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2316, 22bitr3i 264 . . . . . 6 ((((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}) ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2413, 14, 233bitri 284 . . . . 5 ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2512, 24sylib 206 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2625expcom 449 . . 3 (𝑥𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}))
278, 26vtoclga 3244 . 2 (𝐵𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩}))
2827impcom 444 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  wss 3539  {csn 4124  cop 4130  cres 5030   Fn wfn 5785  wf 5786  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798
This theorem is referenced by:  funressn  6309  fressnfv  6310  fnsnsplit  6333  canthp1lem2  9331  fseq1p1m1  12238  dprd2da  18210  dmdprdpr  18217  dprdpr  18218  dpjlem  18219  pgpfaclem1  18249  islindf4  19938  xpstopnlem1  21364  ptcmpfi  21368  2pthlem1  25891  eupath2lem3  26272  subfacp1lem5  30226  cvmliftlem10  30336  poimirlem9  32384  resunimafz0  40188
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