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Theorem fressnfv 6898
Description: The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
fressnfv ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶))

Proof of Theorem fressnfv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 4553 . . . . . 6 (𝑥 = 𝐵 → {𝑥} = {𝐵})
2 reseq2 5824 . . . . . . . 8 ({𝑥} = {𝐵} → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
32feq1d 6475 . . . . . . 7 ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝑥}⟶𝐶))
4 feq2 6472 . . . . . . 7 ({𝑥} = {𝐵} → ((𝐹 ↾ {𝐵}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶))
53, 4bitrd 281 . . . . . 6 ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶))
61, 5syl 17 . . . . 5 (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶))
7 fveq2 6646 . . . . . 6 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
87eleq1d 2895 . . . . 5 (𝑥 = 𝐵 → ((𝐹𝑥) ∈ 𝐶 ↔ (𝐹𝐵) ∈ 𝐶))
96, 8bibi12d 348 . . . 4 (𝑥 = 𝐵 → (((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹𝑥) ∈ 𝐶) ↔ ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶)))
109imbi2d 343 . . 3 (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹𝑥) ∈ 𝐶)) ↔ (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶))))
11 fnressn 6896 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
12 vsnid 4578 . . . . . . . . . 10 𝑥 ∈ {𝑥}
13 fvres 6665 . . . . . . . . . 10 (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥))
1412, 13ax-mp 5 . . . . . . . . 9 ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥)
1514opeq2i 4783 . . . . . . . 8 𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩
1615sneqi 4554 . . . . . . 7 {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹𝑥)⟩}
1716eqeq2i 2833 . . . . . 6 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
18 vex 3476 . . . . . . . 8 𝑥 ∈ V
1918fsn2 6874 . . . . . . 7 ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
20 iba 530 . . . . . . . 8 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩})))
2114eleq1i 2901 . . . . . . . 8 (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (𝐹𝑥) ∈ 𝐶)
2220, 21bitr3di 288 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → ((((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}) ↔ (𝐹𝑥) ∈ 𝐶))
2319, 22syl5bb 285 . . . . . 6 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹𝑥) ∈ 𝐶))
2417, 23sylbir 237 . . . . 5 ((𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹𝑥) ∈ 𝐶))
2511, 24syl 17 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹𝑥) ∈ 𝐶))
2625expcom 416 . . 3 (𝑥𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹𝑥) ∈ 𝐶)))
2710, 26vtoclga 3553 . 2 (𝐵𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶)))
2827impcom 410 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  {csn 4543  cop 4549  cres 5533   Fn wfn 6326  wf 6327  cfv 6331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339
This theorem is referenced by: (None)
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