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Theorem fressnfv 5293
 Description: The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
fressnfv ((𝐹 Fn A B A) → ((𝐹 ↾ {B}):{B}⟶𝐶 ↔ (𝐹B) 𝐶))

Proof of Theorem fressnfv
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sneq 3378 . . . . . 6 (x = B → {x} = {B})
2 reseq2 4550 . . . . . . . 8 ({x} = {B} → (𝐹 ↾ {x}) = (𝐹 ↾ {B}))
32feq1d 4977 . . . . . . 7 ({x} = {B} → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹 ↾ {B}):{x}⟶𝐶))
4 feq2 4974 . . . . . . 7 ({x} = {B} → ((𝐹 ↾ {B}):{x}⟶𝐶 ↔ (𝐹 ↾ {B}):{B}⟶𝐶))
53, 4bitrd 177 . . . . . 6 ({x} = {B} → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹 ↾ {B}):{B}⟶𝐶))
61, 5syl 14 . . . . 5 (x = B → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹 ↾ {B}):{B}⟶𝐶))
7 fveq2 5121 . . . . . 6 (x = B → (𝐹x) = (𝐹B))
87eleq1d 2103 . . . . 5 (x = B → ((𝐹x) 𝐶 ↔ (𝐹B) 𝐶))
96, 8bibi12d 224 . . . 4 (x = B → (((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹x) 𝐶) ↔ ((𝐹 ↾ {B}):{B}⟶𝐶 ↔ (𝐹B) 𝐶)))
109imbi2d 219 . . 3 (x = B → ((𝐹 Fn A → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹x) 𝐶)) ↔ (𝐹 Fn A → ((𝐹 ↾ {B}):{B}⟶𝐶 ↔ (𝐹B) 𝐶))))
11 fnressn 5292 . . . . 5 ((𝐹 Fn A x A) → (𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩})
12 ssnid 3395 . . . . . . . . . 10 x {x}
13 fvres 5141 . . . . . . . . . 10 (x {x} → ((𝐹 ↾ {x})‘x) = (𝐹x))
1412, 13ax-mp 7 . . . . . . . . 9 ((𝐹 ↾ {x})‘x) = (𝐹x)
1514opeq2i 3544 . . . . . . . 8 x, ((𝐹 ↾ {x})‘x)⟩ = ⟨x, (𝐹x)⟩
1615sneqi 3379 . . . . . . 7 {⟨x, ((𝐹 ↾ {x})‘x)⟩} = {⟨x, (𝐹x)⟩}
1716eqeq2i 2047 . . . . . 6 ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} ↔ (𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩})
18 vex 2554 . . . . . . . 8 x V
1918fsn2 5280 . . . . . . 7 ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (((𝐹 ↾ {x})‘x) 𝐶 (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩}))
2014eleq1i 2100 . . . . . . . 8 (((𝐹 ↾ {x})‘x) 𝐶 ↔ (𝐹x) 𝐶)
21 iba 284 . . . . . . . 8 ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} → (((𝐹 ↾ {x})‘x) 𝐶 ↔ (((𝐹 ↾ {x})‘x) 𝐶 (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩})))
2220, 21syl5rbbr 184 . . . . . . 7 ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} → ((((𝐹 ↾ {x})‘x) 𝐶 (𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩}) ↔ (𝐹x) 𝐶))
2319, 22syl5bb 181 . . . . . 6 ((𝐹 ↾ {x}) = {⟨x, ((𝐹 ↾ {x})‘x)⟩} → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹x) 𝐶))
2417, 23sylbir 125 . . . . 5 ((𝐹 ↾ {x}) = {⟨x, (𝐹x)⟩} → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹x) 𝐶))
2511, 24syl 14 . . . 4 ((𝐹 Fn A x A) → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹x) 𝐶))
2625expcom 109 . . 3 (x A → (𝐹 Fn A → ((𝐹 ↾ {x}):{x}⟶𝐶 ↔ (𝐹x) 𝐶)))
2710, 26vtoclga 2613 . 2 (B A → (𝐹 Fn A → ((𝐹 ↾ {B}):{B}⟶𝐶 ↔ (𝐹B) 𝐶)))
2827impcom 116 1 ((𝐹 Fn A B A) → ((𝐹 ↾ {B}):{B}⟶𝐶 ↔ (𝐹B) 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {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-bnd 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: (None)
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