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Theorem refressn 37616
Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 37615) is reflexive, see also refrelressn 37697. (Contributed by Peter Mazsa, 12-Jun-2024.)
Assertion
Ref Expression
refressn (𝐴𝑉 → ∀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
StepHypRef Expression
1 elin 3963 . . . . . 6 (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) ↔ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})))
2 eldmressnALTV 37443 . . . . . . . . 9 (𝑥 ∈ V → (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴𝐴 ∈ dom 𝑅)))
32elv 3478 . . . . . . . 8 (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴𝐴 ∈ dom 𝑅))
43simplbi 496 . . . . . . 7 (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝑥 = 𝐴)
54adantr 479 . . . . . 6 ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → 𝑥 = 𝐴)
61, 5sylbi 216 . . . . 5 (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥 = 𝐴)
76a1i 11 . . . 4 (𝐴𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥 = 𝐴))
8 elrnressn 37444 . . . . . . . . 9 ((𝐴𝑉𝑥 ∈ V) → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
98elvd 3479 . . . . . . . 8 (𝐴𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
109biimpd 228 . . . . . . 7 (𝐴𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) → 𝐴𝑅𝑥))
1110adantld 489 . . . . . 6 (𝐴𝑉 → ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
121, 11biimtrid 241 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
134eqcomd 2736 . . . . . . . 8 (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝐴 = 𝑥)
1413breq1d 5157 . . . . . . 7 (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → (𝐴𝑅𝑥𝑥𝑅𝑥))
1514adantr 479 . . . . . 6 ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥𝑥𝑅𝑥))
161, 15sylbi 216 . . . . 5 (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥𝑥𝑅𝑥))
1712, 16mpbidi 240 . . . 4 (𝐴𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥𝑅𝑥))
187, 17jcad 511 . . 3 (𝐴𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝑥 = 𝐴𝑥𝑅𝑥)))
19 brressn 37614 . . . 4 ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴𝑥𝑅𝑥)))
2019el2v 3480 . . 3 (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴𝑥𝑅𝑥))
2118, 20imbitrrdi 251 . 2 (𝐴𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥(𝑅 ↾ {𝐴})𝑥))
2221ralrimiv 3143 1 (𝐴𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wral 3059  Vcvv 3472  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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  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  37697
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