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Theorem fressnfv 6467
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 4220 . . . . . 6 (𝑥 = 𝐵 → {𝑥} = {𝐵})
2 reseq2 5423 . . . . . . . 8 ({𝑥} = {𝐵} → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
32feq1d 6068 . . . . . . 7 ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝑥}⟶𝐶))
4 feq2 6065 . . . . . . 7 ({𝑥} = {𝐵} → ((𝐹 ↾ {𝐵}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶))
53, 4bitrd 268 . . . . . 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 6229 . . . . . 6 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
87eleq1d 2715 . . . . 5 (𝑥 = 𝐵 → ((𝐹𝑥) ∈ 𝐶 ↔ (𝐹𝐵) ∈ 𝐶))
96, 8bibi12d 334 . . . 4 (𝑥 = 𝐵 → (((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹𝑥) ∈ 𝐶) ↔ ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶)))
109imbi2d 329 . . 3 (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹𝑥) ∈ 𝐶)) ↔ (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶))))
11 fnressn 6465 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
12 vsnid 4242 . . . . . . . . . 10 𝑥 ∈ {𝑥}
13 fvres 6245 . . . . . . . . . 10 (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥))
1412, 13ax-mp 5 . . . . . . . . 9 ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥)
1514opeq2i 4437 . . . . . . . 8 𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩
1615sneqi 4221 . . . . . . 7 {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹𝑥)⟩}
1716eqeq2i 2663 . . . . . 6 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
18 vex 3234 . . . . . . . 8 𝑥 ∈ V
1918fsn2 6443 . . . . . . 7 ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
2014eleq1i 2721 . . . . . . . 8 (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (𝐹𝑥) ∈ 𝐶)
21 iba 523 . . . . . . . 8 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩})))
2220, 21syl5rbbr 275 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → ((((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}) ↔ (𝐹𝑥) ∈ 𝐶))
2319, 22syl5bb 272 . . . . . 6 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹𝑥) ∈ 𝐶))
2417, 23sylbir 225 . . . . 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 450 . . 3 (𝑥𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹𝑥) ∈ 𝐶)))
2710, 26vtoclga 3303 . 2 (𝐵𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶)))
2827impcom 445 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹𝐵) ∈ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {csn 4210  cop 4216  cres 5145   Fn wfn 5921  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by: (None)
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