Theorem refressn 38706

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38705) is reflexive, see also refrelressn 38777. (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 3917	. . . . . 6 ⊢ (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) ↔ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})))
2		eldmressnALTV 38472	. . . . . . . . 9 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝐴 ∈ dom 𝑅)))
3	2	elv 3445	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝐴 ∈ dom 𝑅))
4	3	simplbi 497	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝑥 = 𝐴)
5	4	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → 𝑥 = 𝐴)
6	1, 5	sylbi 217	. . . . 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 38473	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
9	8	elvd 3446	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
10	9	biimpd 229	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) → 𝐴𝑅𝑥))
11	10	adantld 490	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
12	1, 11	biimtrid 242	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
13	4	eqcomd 2742	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝐴 = 𝑥)
14	13	breq1d 5108	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
15	14	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
16	1, 15	sylbi 217	. . . . 5 ⊢ (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
17	12, 16	mpbidi 241	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥𝑅𝑥))
18	7, 17	jcad 512	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥)))
19		brressn 38704	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥)))
20	19	el2v 3447	. . 3 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥))
21	18, 20	imbitrrdi 252	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥(𝑅 ↾ {𝐴})𝑥))
22	21	ralrimiv 3127	1 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ∩ cin 3900  {csn 4580   class class class wbr 5098  dom cdm 5624  ran crn 5625   ↾ cres 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636

This theorem is referenced by:  refrelressn  38777