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Theorem fnressn 5292
Description: A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
fnressn ((𝐹 Fn A B A) → (𝐹 ↾ {B}) = {⟨B, (𝐹B)⟩})

Proof of Theorem fnressn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sneq 3378 . . . . . 6 (x = B → {x} = {B})
21reseq2d 4555 . . . . 5 (x = B → (𝐹 ↾ {x}) = (𝐹 ↾ {B}))
3 fveq2 5121 . . . . . . 7 (x = B → (𝐹x) = (𝐹B))
4 opeq12 3542 . . . . . . 7 ((x = B (𝐹x) = (𝐹B)) → ⟨x, (𝐹x)⟩ = ⟨B, (𝐹B)⟩)
53, 4mpdan 398 . . . . . 6 (x = B → ⟨x, (𝐹x)⟩ = ⟨B, (𝐹B)⟩)
65sneqd 3380 . . . . 5 (x = B → {⟨x, (𝐹x)⟩} = {⟨B, (𝐹B)⟩})
72, 6eqeq12d 2051 . . . 4 (x = B → ((𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩} ↔ (𝐹 ↾ {B}) = {⟨B, (𝐹B)⟩}))
87imbi2d 219 . . 3 (x = B → ((𝐹 Fn A → (𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩}) ↔ (𝐹 Fn A → (𝐹 ↾ {B}) = {⟨B, (𝐹B)⟩})))
9 vex 2554 . . . . . . 7 x V
109snss 3485 . . . . . 6 (x A ↔ {x} ⊆ A)
11 fnssres 4955 . . . . . 6 ((𝐹 Fn A {x} ⊆ A) → (𝐹 ↾ {x}) Fn {x})
1210, 11sylan2b 271 . . . . 5 ((𝐹 Fn A x A) → (𝐹 ↾ {x}) Fn {x})
13 dffn2 4990 . . . . . . 7 ((𝐹 ↾ {x}) Fn {x} ↔ (𝐹 ↾ {x}):{x}⟶V)
149fsn2 5280 . . . . . . 7 ((𝐹 ↾ {x}):{x}⟶V ↔ (((𝐹 ↾ {x})‘x) V (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩}))
1513, 14bitri 173 . . . . . 6 ((𝐹 ↾ {x}) Fn {x} ↔ (((𝐹 ↾ {x})‘x) V (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩}))
16 ssnid 3395 . . . . . . . . . . 11 x {x}
17 fvres 5141 . . . . . . . . . . 11 (x {x} → ((𝐹 ↾ {x})‘x) = (𝐹x))
1816, 17ax-mp 7 . . . . . . . . . 10 ((𝐹 ↾ {x})‘x) = (𝐹x)
1918opeq2i 3544 . . . . . . . . 9 x, ((𝐹 ↾ {x})‘x)⟩ = ⟨x, (𝐹x)⟩
2019sneqi 3379 . . . . . . . 8 {⟨x, ((𝐹 ↾ {x})‘x)⟩} = {⟨x, (𝐹x)⟩}
2120eqeq2i 2047 . . . . . . 7 ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} ↔ (𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩})
22 snssi 3499 . . . . . . . . . 10 (x A → {x} ⊆ A)
2322, 11sylan2 270 . . . . . . . . 9 ((𝐹 Fn A x A) → (𝐹 ↾ {x}) Fn {x})
24 funfvex 5135 . . . . . . . . . 10 ((Fun (𝐹 ↾ {x}) x dom (𝐹 ↾ {x})) → ((𝐹 ↾ {x})‘x) V)
2524funfni 4942 . . . . . . . . 9 (((𝐹 ↾ {x}) Fn {x} x {x}) → ((𝐹 ↾ {x})‘x) V)
2623, 16, 25sylancl 392 . . . . . . . 8 ((𝐹 Fn A x A) → ((𝐹 ↾ {x})‘x) V)
2726biantrurd 289 . . . . . . 7 ((𝐹 Fn A x A) → ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} ↔ (((𝐹 ↾ {x})‘x) V (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩})))
2821, 27syl5rbbr 184 . . . . . 6 ((𝐹 Fn A x A) → ((((𝐹 ↾ {x})‘x) V (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩}) ↔ (𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩}))
2915, 28syl5bb 181 . . . . 5 ((𝐹 Fn A x A) → ((𝐹 ↾ {x}) Fn {x} ↔ (𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩}))
3012, 29mpbid 135 . . . 4 ((𝐹 Fn A x A) → (𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩})
3130expcom 109 . . 3 (x A → (𝐹 Fn A → (𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩}))
328, 31vtoclga 2613 . 2 (B A → (𝐹 Fn A → (𝐹 ↾ {B}) = {⟨B, (𝐹B)⟩}))
3332impcom 116 1 ((𝐹 Fn A B A) → (𝐹 ↾ {B}) = {⟨B, (𝐹B)⟩})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  wss 2911  {csn 3367  cop 3370  cres 4290   Fn wfn 4840  wf 4841  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by:  fressnfv  5293  fseq1p1m1  8726
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