Theorem refressn 37313

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 37312) is reflexive, see also refrelressn 37394. (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 3965	. . . . . 6 ⊢ (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) ↔ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})))
2		eldmressnALTV 37140	. . . . . . . . 9 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝐴 ∈ dom 𝑅)))
3	2	elv 3481	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝐴 ∈ dom 𝑅))
4	3	simplbi 499	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝑥 = 𝐴)
5	4	adantr 482	. . . . . 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 37141	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
9	8	elvd 3482	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
10	9	biimpd 228	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) → 𝐴𝑅𝑥))
11	10	adantld 492	. . . . . 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 2739	. . . . . . . 8 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝐴 = 𝑥)
14	13	breq1d 5159	. . . . . . 7 ⊢ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → (𝐴𝑅𝑥 ↔ 𝑥𝑅𝑥))
15	14	adantr 482	. . . . . 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 514	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥)))
19		brressn 37311	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥)))
20	19	el2v 3483	. . 3 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑥))
21	18, 20	syl6ibr 252	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥(𝑅 ↾ {𝐴})𝑥))
22	21	ralrimiv 3146	1 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ∩ cin 3948  {csn 4629   class class class wbr 5149  dom cdm 5677  ran crn 5678   ↾ cres 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689

This theorem is referenced by:  refrelressn  37394