Theorem dfdfat2 44620

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.

Step	Hyp	Ref	Expression
1		df-dfat 44611	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		relres 5920	. . . 4 ⊢ Rel (𝐹 ↾ {𝐴})
3		dffun8 6462	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
4	2, 3	mpbiran 706	. . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
5	4	anbi2i 623	. 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
6		brres 5898	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
7	6	elv 3438	. . . . . . 7 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦))
8	7	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
9	8	eubidv 2586	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → (∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
10	9	ralbidv 3112	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
11		eldmressnsn 5934	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))
12		eldmressn 44531	. . . . 5 ⊢ (𝑥 ∈ dom (𝐹 ↾ {𝐴}) → 𝑥 = 𝐴)
13		velsn 4577	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
14	13	biimpri 227	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑥 ∈ {𝐴})
15		breq1 5077	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
16	15	anbi2d 629	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ {𝐴} ∧ 𝐴𝐹𝑦)))
17	14, 16	mpbirand 704	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ 𝐴𝐹𝑦))
18	17	eubidv 2586	. . . . 5 ⊢ (𝑥 = 𝐴 → (∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ ∃!𝑦 𝐴𝐹𝑦))
19	11, 12, 18	ralbinrald 44614	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ ∃!𝑦 𝐴𝐹𝑦))
20	10, 19	bitrd 278	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
21	20	pm5.32i 575	. 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
22	1, 5, 21	3bitri 297	1 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃!weu 2568  ∀wral 3064  Vcvv 3432  {csn 4561   class class class wbr 5074  dom cdm 5589   ↾ cres 5591  Rel wrel 5594  Fun wfun 6427   defAt wdfat 44608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-fun 6435  df-dfat 44611

This theorem is referenced by:  dfafv2  44624  afveu  44645  rlimdmafv  44669  tz6.12-2-afv2  44729  afv2eu  44730  tz6.12i-afv2  44735  dfatbrafv2b  44737  dfatsnafv2  44744  dfafv23  44745  dfatcolem  44747  dfatco  44748  rlimdmafv2  44750