Theorem refressn 38900

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38899) is reflexive, see also refrelressn 38971. (Contributed by Peter Mazsa, 12-Jun-2024.)

Assertion

Ref	Expression
refressn	⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hint:   𝑅(𝑥)

Proof of Theorem refressn

Step	Hyp	Ref	Expression
1		elin 3899	. . . . . 6 ⊢ (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) ↔ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})))
2		eldmressnALTV 38646	. . . . . . . . 9 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝐴 ∈ dom 𝑅)))
3	2	elv 3436	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝐴 ∈ dom 𝑅))
4	3	simplbi 497	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝑥 = 𝐴)
5	4	adantr 481	. . . . . 6 ⊢ ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → 𝑥 = 𝐴)
6	1, 5	sylbi 218	. . . . 5 ⊢ (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥 = 𝐴)
7	6	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥 = 𝐴))
8		elrnressn 38647	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
9	8	elvd 3437	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
10	9	biimpd 230	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) → 𝐴𝑅𝑥))
11	10	adantld 491	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
12	1, 11	biimtrid 243	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
13	4	eqcomd 2745	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝐴 = 𝑥)
14	13	breq1d 5082	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
15	14	adantr 481	. . . . . 6 ⊢ ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
16	1, 15	sylbi 218	. . . . 5 ⊢ (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
17	12, 16	mpbidi 242	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥𝑅𝑥))
18	7, 17	jcad 517	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥)))
19		brressn 38898	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥)))
20	19	el2v 3438	. . 3 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥))
21	18, 20	imbitrrdi 253	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥(𝑅 ↾ {𝐴})𝑥))
22	21	ralrimiv 3130	1 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ∩ cin 3882  {csn 4555   class class class wbr 5072  dom cdm 5618  ran crn 5619   ↾ cres 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630

This theorem is referenced by:  refrelressn  38971