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Theorem fnressn 5270
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 3357 . . . . . 6 (x = B → {x} = {B})
21reseq2d 4535 . . . . 5 (x = B → (𝐹 ↾ {x}) = (𝐹 ↾ {B}))
3 fveq2 5099 . . . . . . 7 (x = B → (𝐹x) = (𝐹B))
4 opeq12 3521 . . . . . . 7 ((x = B (𝐹x) = (𝐹B)) → ⟨x, (𝐹x)⟩ = ⟨B, (𝐹B)⟩)
53, 4mpdan 400 . . . . . 6 (x = B → ⟨x, (𝐹x)⟩ = ⟨B, (𝐹B)⟩)
65sneqd 3359 . . . . 5 (x = B → {⟨x, (𝐹x)⟩} = {⟨B, (𝐹B)⟩})
72, 6eqeq12d 2032 . . . 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 2534 . . . . . . 7 x V
109snss 3464 . . . . . 6 (x A ↔ {x} ⊆ A)
11 fnssres 4934 . . . . . 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 4969 . . . . . . 7 ((𝐹 ↾ {x}) Fn {x} ↔ (𝐹 ↾ {x}):{x}⟶V)
149fsn2 5258 . . . . . . 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 3374 . . . . . . . . . . 11 x {x}
17 fvres 5119 . . . . . . . . . . 11 (x {x} → ((𝐹 ↾ {x})‘x) = (𝐹x))
1816, 17ax-mp 7 . . . . . . . . . 10 ((𝐹 ↾ {x})‘x) = (𝐹x)
1918opeq2i 3523 . . . . . . . . 9 x, ((𝐹 ↾ {x})‘x)⟩ = ⟨x, (𝐹x)⟩
2019sneqi 3358 . . . . . . . 8 {⟨x, ((𝐹 ↾ {x})‘x)⟩} = {⟨x, (𝐹x)⟩}
2120eqeq2i 2028 . . . . . . 7 ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} ↔ (𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩})
22 snssi 3478 . . . . . . . . . 10 (x A → {x} ⊆ A)
2322, 11sylan2 270 . . . . . . . . 9 ((𝐹 Fn A x A) → (𝐹 ↾ {x}) Fn {x})
24 funfvex 5113 . . . . . . . . . 10 ((Fun (𝐹 ↾ {x}) x dom (𝐹 ↾ {x})) → ((𝐹 ↾ {x})‘x) V)
2524funfni 4921 . . . . . . . . 9 (((𝐹 ↾ {x}) Fn {x} x {x}) → ((𝐹 ↾ {x})‘x) V)
2623, 16, 25sylancl 394 . . . . . . . 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 2592 . 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 1226   wcel 1370  Vcvv 2531  wss 2890  {csn 3346  cop 3349  cres 4270   Fn wfn 4820  wf 4821  cfv 4825
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-reu 2287  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833
This theorem is referenced by:  fressnfv  5271
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