Theorem dfdfat2 47078

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 47069	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		relres 6026	. . . 4 ⊢ Rel (𝐹 ↾ {𝐴})
3		dffun8 6596	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
4	2, 3	mpbiran 709	. . 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 6007	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
7	6	elv 3483	. . . . . . 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 2584	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → (∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
10	9	ralbidv 3176	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
11		eldmressnsn 6044	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))
12		eldmressn 46987	. . . . 5 ⊢ (𝑥 ∈ dom (𝐹 ↾ {𝐴}) → 𝑥 = 𝐴)
13		velsn 4647	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
14	13	biimpri 228	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑥 ∈ {𝐴})
15		breq1 5151	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
16	15	anbi2d 630	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ {𝐴} ∧ 𝐴𝐹𝑦)))
17	14, 16	mpbirand 707	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ 𝐴𝐹𝑦))
18	17	eubidv 2584	. . . . 5 ⊢ (𝑥 = 𝐴 → (∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ ∃!𝑦 𝐴𝐹𝑦))
19	11, 12, 18	ralbinrald 47072	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ ∃!𝑦 𝐴𝐹𝑦))
20	10, 19	bitrd 279	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
21	20	pm5.32i 574	. 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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃!weu 2566  ∀wral 3059  Vcvv 3478  {csn 4631   class class class wbr 5148  dom cdm 5689   ↾ cres 5691  Rel wrel 5694  Fun wfun 6557   defAt wdfat 47066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-fun 6565  df-dfat 47069

This theorem is referenced by:  dfafv2  47082  afveu  47103  rlimdmafv  47127  tz6.12-2-afv2  47187  afv2eu  47188  tz6.12i-afv2  47193  dfatbrafv2b  47195  dfatsnafv2  47202  dfafv23  47203  dfatcolem  47205  dfatco  47206  rlimdmafv2  47208