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Theorem dfdfat2 47145
Description: Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
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 47136 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 relres 6022 . . . 4 Rel (𝐹 ↾ {𝐴})
3 dffun8 6593 . . . 4 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
42, 3mpbiran 709 . . 3 (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
54anbi2i 623 . 2 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
6 brres 6003 . . . . . . . 8 (𝑦 ∈ V → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
76elv 3484 . . . . . . 7 (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦))
87a1i 11 . . . . . 6 (𝐴 ∈ dom 𝐹 → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
98eubidv 2585 . . . . 5 (𝐴 ∈ dom 𝐹 → (∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
109ralbidv 3177 . . . 4 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
11 eldmressnsn 6041 . . . . 5 (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹 ↾ {𝐴}))
12 eldmressn 47054 . . . . 5 (𝑥 ∈ dom (𝐹 ↾ {𝐴}) → 𝑥 = 𝐴)
13 velsn 4641 . . . . . . . 8 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
1413biimpri 228 . . . . . . 7 (𝑥 = 𝐴𝑥 ∈ {𝐴})
15 breq1 5145 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
1615anbi2d 630 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ {𝐴} ∧ 𝐴𝐹𝑦)))
1714, 16mpbirand 707 . . . . . 6 (𝑥 = 𝐴 → ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ 𝐴𝐹𝑦))
1817eubidv 2585 . . . . 5 (𝑥 = 𝐴 → (∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ ∃!𝑦 𝐴𝐹𝑦))
1911, 12, 18ralbinrald 47139 . . . 4 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ ∃!𝑦 𝐴𝐹𝑦))
2010, 19bitrd 279 . . 3 (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
2120pm5.32i 574 . 2 ((𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
221, 5, 213bitri 297 1 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  ∃!weu 2567  wral 3060  Vcvv 3479  {csn 4625   class class class wbr 5142  dom cdm 5684  cres 5686  Rel wrel 5689  Fun wfun 6554   defAt wdfat 47133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-fun 6562  df-dfat 47136
This theorem is referenced by:  dfafv2  47149  afveu  47170  rlimdmafv  47194  tz6.12-2-afv2  47254  afv2eu  47255  tz6.12i-afv2  47260  dfatbrafv2b  47262  dfatsnafv2  47269  dfafv23  47270  dfatcolem  47272  dfatco  47273  rlimdmafv2  47275
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