Theorem dfdfat2 47719

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 47710	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		relres 5991	. . . 4 ⊢ Rel (𝐹 ↾ {𝐴})
3		dffun8 6549	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
4	2, 3	mpbiran 719	. . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
5	4	anbi2i 632	. 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
6		brres 5972	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
7	6	elv 3459	. . . . . . 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 2613	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → (∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
10	9	ralbidv 3185	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦)))
11		eldmressnsn 6010	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))
12		eldmressn 47628	. . . . 5 ⊢ (𝑥 ∈ dom (𝐹 ↾ {𝐴}) → 𝑥 = 𝐴)
13		velsn 4598	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
14	13	biimpri 230	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑥 ∈ {𝐴})
15		breq1 5103	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
16	15	anbi2d 639	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ {𝐴} ∧ 𝐴𝐹𝑦)))
17	14, 16	mpbirand 717	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ 𝐴𝐹𝑦))
18	17	eubidv 2613	. . . . 5 ⊢ (𝑥 = 𝐴 → (∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ ∃!𝑦 𝐴𝐹𝑦))
19	11, 12, 18	ralbinrald 47713	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) ↔ ∃!𝑦 𝐴𝐹𝑦))
20	10, 19	bitrd 281	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
21	20	pm5.32i 582	. 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
22	1, 5, 21	3bitri 299	1 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃!weu 2595  ∀wral 3076  Vcvv 3454  {csn 4582   class class class wbr 5100  dom cdm 5647   ↾ cres 5649  Rel wrel 5652  Fun wfun 6515   defAt wdfat 47707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-res 5659  df-fun 6523  df-dfat 47710

This theorem is referenced by:  dfafv2  47723  afveu  47744  rlimdmafv  47768  tz6.12-2-afv2  47828  afv2eu  47829  tz6.12i-afv2  47834  dfatbrafv2b  47836  dfatsnafv2  47843  dfafv23  47844  dfatcolem  47846  dfatco  47847  rlimdmafv2  47849