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Theorem dfdfat2 39860
Description: Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
Assertion
Ref Expression
dfdfat2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem dfdfat2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-dfat 39845 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 relres 5346 . . . 4 Rel (𝐹 ↾ {𝐴})
3 dffun8 5831 . . . 4 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
42, 3mpbiran 955 . . 3 (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
54anbi2i 726 . 2 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
6 vex 3176 . . . . . . . 8 𝑦 ∈ V
76brres 5323 . . . . . . 7 (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝐴}))
87a1i 11 . . . . . 6 (𝐴 ∈ dom 𝐹 → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝐴})))
98eubidv 2478 . . . . 5 (𝐴 ∈ dom 𝐹 → (∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴})))
109ralbidv 2969 . . . 4 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴})))
11 eldmressnsn 5359 . . . . 5 (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹 ↾ {𝐴}))
12 eldmressn 39852 . . . . 5 (𝑥 ∈ dom (𝐹 ↾ {𝐴}) → 𝑥 = 𝐴)
13 breq1 4586 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
1413anbi1d 737 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ (𝐴𝐹𝑦𝑥 ∈ {𝐴})))
15 velsn 4141 . . . . . . . . 9 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
1615biimpri 217 . . . . . . . 8 (𝑥 = 𝐴𝑥 ∈ {𝐴})
1716biantrud 527 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝐹𝑦 ↔ (𝐴𝐹𝑦𝑥 ∈ {𝐴})))
1814, 17bitr4d 270 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ 𝐴𝐹𝑦))
1918eubidv 2478 . . . . 5 (𝑥 = 𝐴 → (∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ ∃!𝑦 𝐴𝐹𝑦))
2011, 12, 19ralbinrald 39848 . . . 4 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ ∃!𝑦 𝐴𝐹𝑦))
2110, 20bitrd 267 . . 3 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
2221pm5.32i 667 . 2 ((𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
231, 5, 223bitri 285 1 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  ∃!weu 2458  wral 2896  {csn 4125   class class class wbr 4583  dom cdm 5038  cres 5040  Rel wrel 5043  Fun wfun 5798   defAt wdfat 39842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-fun 5806  df-dfat 39845
This theorem is referenced by:  afveu  39882  rlimdmafv  39906
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