Theorem refressn 39182

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 39181) is reflexive, see also refrelressn 39253. (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 3921	. . . . . 6 ⊢ (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) ↔ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})))
2		eldmressnALTV 38928	. . . . . . . . 9 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝐴 ∈ dom 𝑅)))
3	2	elv 3460	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝐴 ∈ dom 𝑅))
4	3	simplbi 501	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝑥 = 𝐴)
5	4	adantr 485	. . . . . 6 ⊢ ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → 𝑥 = 𝐴)
6	1, 5	sylbi 220	. . . . 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 38929	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
9	8	elvd 3461	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
10	9	biimpd 232	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) → 𝐴𝑅𝑥))
11	10	adantld 495	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
12	1, 11	biimtrid 245	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
13	4	eqcomd 2769	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝐴 = 𝑥)
14	13	breq1d 5119	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
15	14	adantr 485	. . . . . 6 ⊢ ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
16	1, 15	sylbi 220	. . . . 5 ⊢ (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
17	12, 16	mpbidi 244	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥𝑅𝑥))
18	7, 17	jcad 521	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥)))
19		brressn 39180	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥)))
20	19	el2v 3462	. . 3 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥))
21	18, 20	imbitrrdi 255	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥(𝑅 ↾ {𝐴})𝑥))
22	21	ralrimiv 3156	1 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∩ cin 3904  {csn 4589   class class class wbr 5109  dom cdm 5661  ran crn 5662   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673

This theorem is referenced by:  refrelressn  39253