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Theorem dfdfat2 41532
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 41517 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 relres 5461 . . . 4 Rel (𝐹 ↾ {𝐴})
3 dffun8 5954 . . . 4 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
42, 3mpbiran 973 . . 3 (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
54anbi2i 730 . 2 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
6 vex 3234 . . . . . . . 8 𝑦 ∈ V
76brres 5437 . . . . . . 7 (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝐴}))
87a1i 11 . . . . . 6 (𝐴 ∈ dom 𝐹 → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝐴})))
98eubidv 2518 . . . . 5 (𝐴 ∈ dom 𝐹 → (∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴})))
109ralbidv 3015 . . . 4 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴})))
11 eldmressnsn 5474 . . . . 5 (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹 ↾ {𝐴}))
12 eldmressn 41524 . . . . 5 (𝑥 ∈ dom (𝐹 ↾ {𝐴}) → 𝑥 = 𝐴)
13 breq1 4688 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
1413anbi1d 741 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ (𝐴𝐹𝑦𝑥 ∈ {𝐴})))
15 velsn 4226 . . . . . . . . 9 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
1615biimpri 218 . . . . . . . 8 (𝑥 = 𝐴𝑥 ∈ {𝐴})
1716biantrud 527 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝐹𝑦 ↔ (𝐴𝐹𝑦𝑥 ∈ {𝐴})))
1814, 17bitr4d 271 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ 𝐴𝐹𝑦))
1918eubidv 2518 . . . . 5 (𝑥 = 𝐴 → (∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ ∃!𝑦 𝐴𝐹𝑦))
2011, 12, 19ralbinrald 41520 . . . 4 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥𝐹𝑦𝑥 ∈ {𝐴}) ↔ ∃!𝑦 𝐴𝐹𝑦))
2110, 20bitrd 268 . . 3 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
2221pm5.32i 670 . 2 ((𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
231, 5, 223bitri 286 1 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  ∃!weu 2498  wral 2941  {csn 4210   class class class wbr 4685  dom cdm 5143  cres 5145  Rel wrel 5148  Fun wfun 5920   defAt wdfat 41514
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-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  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-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-fun 5928  df-dfat 41517
This theorem is referenced by:  afveu  41554  rlimdmafv  41578
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