Theorem refressn 37301

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 37300) is reflexive, see also refrelressn 37382. (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 3963	. . . . . 6 ⊢ (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) ↔ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})))
2		eldmressnALTV 37128	. . . . . . . . 9 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝐴 ∈ dom 𝑅)))
3	2	elv 3480	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝐴 ∈ dom 𝑅))
4	3	simplbi 498	. . . . . . 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 216	. . . . 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 37129	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
9	8	elvd 3481	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
10	9	biimpd 228	. . . . . . 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 241	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
13	4	eqcomd 2738	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝐴 = 𝑥)
14	13	breq1d 5157	. . . . . . 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 216	. . . . 5 ⊢ (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
17	12, 16	mpbidi 240	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥𝑅𝑥))
18	7, 17	jcad 513	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥)))
19		brressn 37299	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥)))
20	19	el2v 3482	. . 3 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥))
21	18, 20	syl6ibr 251	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥(𝑅 ↾ {𝐴})𝑥))
22	21	ralrimiv 3145	1 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ∩ cin 3946  {csn 4627   class class class wbr 5147  dom cdm 5675  ran crn 5676   ↾ cres 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687

This theorem is referenced by:  refrelressn  37382