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Theorem fnressn 5702
Description: A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
fnressn ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})

Proof of Theorem fnressn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 3603 . . . . . 6 (𝑥 = 𝐵 → {𝑥} = {𝐵})
21reseq2d 4907 . . . . 5 (𝑥 = 𝐵 → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵}))
3 fveq2 5515 . . . . . . 7 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
4 opeq12 3780 . . . . . . 7 ((𝑥 = 𝐵 ∧ (𝐹𝑥) = (𝐹𝐵)) → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
53, 4mpdan 421 . . . . . 6 (𝑥 = 𝐵 → ⟨𝑥, (𝐹𝑥)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
65sneqd 3605 . . . . 5 (𝑥 = 𝐵 → {⟨𝑥, (𝐹𝑥)⟩} = {⟨𝐵, (𝐹𝐵)⟩})
72, 6eqeq12d 2192 . . . 4 (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩} ↔ (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩}))
87imbi2d 230 . . 3 (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}) ↔ (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})))
9 vex 2740 . . . . . . 7 𝑥 ∈ V
109snss 3727 . . . . . 6 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
11 fnssres 5329 . . . . . 6 ((𝐹 Fn 𝐴 ∧ {𝑥} ⊆ 𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
1210, 11sylan2b 287 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
13 dffn2 5367 . . . . . . 7 ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}):{𝑥}⟶V)
149fsn2 5690 . . . . . . 7 ((𝐹 ↾ {𝑥}):{𝑥}⟶V ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
1513, 14bitri 184 . . . . . 6 ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}))
16 snssi 3736 . . . . . . . . . 10 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
1716, 11sylan2 286 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) Fn {𝑥})
18 vsnid 3624 . . . . . . . . 9 𝑥 ∈ {𝑥}
19 funfvex 5532 . . . . . . . . . 10 ((Fun (𝐹 ↾ {𝑥}) ∧ 𝑥 ∈ dom (𝐹 ↾ {𝑥})) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
2019funfni 5316 . . . . . . . . 9 (((𝐹 ↾ {𝑥}) Fn {𝑥} ∧ 𝑥 ∈ {𝑥}) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
2117, 18, 20sylancl 413 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹 ↾ {𝑥})‘𝑥) ∈ V)
2221biantrurd 305 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩})))
23 fvres 5539 . . . . . . . . . . 11 (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥))
2418, 23ax-mp 5 . . . . . . . . . 10 ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹𝑥)
2524opeq2i 3782 . . . . . . . . 9 𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩
2625sneqi 3604 . . . . . . . 8 {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} = {⟨𝑥, (𝐹𝑥)⟩}
2726eqeq2i 2188 . . . . . . 7 ((𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
2822, 27bitr3di 195 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → ((((𝐹 ↾ {𝑥})‘𝑥) ∈ V ∧ (𝐹 ↾ {𝑥}) = {⟨𝑥, ((𝐹 ↾ {𝑥})‘𝑥)⟩}) ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}))
2915, 28bitrid 192 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹 ↾ {𝑥}) Fn {𝑥} ↔ (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}))
3012, 29mpbid 147 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩})
3130expcom 116 . . 3 (𝑥𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹𝑥)⟩}))
328, 31vtoclga 2803 . 2 (𝐵𝐴 → (𝐹 Fn 𝐴 → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩}))
3332impcom 125 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) = {⟨𝐵, (𝐹𝐵)⟩})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  wss 3129  {csn 3592  cop 3595  cres 4628   Fn wfn 5211  wf 5212  cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224
This theorem is referenced by:  fressnfv  5703  fnsnsplitss  5715  fnsnsplitdc  6505  dif1en  6878  fnfi  6935  fseq1p1m1  10091  resunimafz0  10806
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