Theorem dfdfat2 41211

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.

Step	Hyp	Ref	Expression
1		df-dfat 41196	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		relres 5426	. . . 4 ⊢ Rel (𝐹 ↾ {𝐴})
3		dffun8 5916	. . . 4 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
4	2, 3	mpbiran 953	. . 3 ⊢ (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
5	4	anbi2i 730	. 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
6		vex 3203	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	6	brres 5402	. . . . . . 7 ⊢ (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}))
8	7	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 → (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴})))
9	8	eubidv 2490	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → (∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦(𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴})))
10	9	ralbidv 2986	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴})))
11		eldmressnsn 5439	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))
12		eldmressn 41203	. . . . 5 ⊢ (𝑥 ∈ dom (𝐹 ↾ {𝐴}) → 𝑥 = 𝐴)
13		breq1 4656	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
14	13	anbi1d 741	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ (𝐴𝐹𝑦 ∧ 𝑥 ∈ {𝐴})))
15		velsn 4193	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
16	15	biimpri 218	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝑥 ∈ {𝐴})
17	16	biantrud 528	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴𝐹𝑦 ↔ (𝐴𝐹𝑦 ∧ 𝑥 ∈ {𝐴})))
18	14, 17	bitr4d 271	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ 𝐴𝐹𝑦))
19	18	eubidv 2490	. . . . 5 ⊢ (𝑥 = 𝐴 → (∃!𝑦(𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ ∃!𝑦 𝐴𝐹𝑦))
20	11, 12, 19	ralbinrald 41199	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦(𝑥𝐹𝑦 ∧ 𝑥 ∈ {𝐴}) ↔ ∃!𝑦 𝐴𝐹𝑦))
21	10, 20	bitrd 268	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → (∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
22	21	pm5.32i 669	. 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∀𝑥 ∈ dom (𝐹 ↾ {𝐴})∃!𝑦 𝑥(𝐹 ↾ {𝐴})𝑦) ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
23	1, 5, 22	3bitri 286	1 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃!weu 2470  ∀wral 2912  {csn 4177   class class class wbr 4653  dom cdm 5114   ↾ cres 5116  Rel wrel 5119  Fun wfun 5882   defAt wdfat 41193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-fun 5890  df-dfat 41196

This theorem is referenced by:  afveu  41233  rlimdmafv  41257