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Theorem refressn 38424
Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38423) is reflexive, see also refrelressn 38505. (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 3978 . . . . . 6 (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) ↔ (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})))
2 eldmressnALTV 38253 . . . . . . . . 9 (𝑥 ∈ V → (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴𝐴 ∈ dom 𝑅)))
32elv 3482 . . . . . . . 8 (𝑥 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝑥 = 𝐴𝐴 ∈ dom 𝑅))
43simplbi 497 . . . . . . 7 (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝑥 = 𝐴)
54adantr 480 . . . . . 6 ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → 𝑥 = 𝐴)
61, 5sylbi 217 . . . . 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 38254 . . . . . . . . 9 ((𝐴𝑉𝑥 ∈ V) → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
98elvd 3483 . . . . . . . 8 (𝐴𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝑥))
109biimpd 229 . . . . . . 7 (𝐴𝑉 → (𝑥 ∈ ran (𝑅 ↾ {𝐴}) → 𝐴𝑅𝑥))
1110adantld 490 . . . . . 6 (𝐴𝑉 → ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
121, 11biimtrid 242 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝐴𝑅𝑥))
134eqcomd 2740 . . . . . . . 8 (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → 𝐴 = 𝑥)
1413breq1d 5157 . . . . . . 7 (𝑥 ∈ dom (𝑅 ↾ {𝐴}) → (𝐴𝑅𝑥𝑥𝑅𝑥))
1514adantr 480 . . . . . 6 ((𝑥 ∈ dom (𝑅 ↾ {𝐴}) ∧ 𝑥 ∈ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥𝑥𝑅𝑥))
161, 15sylbi 217 . . . . 5 (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝐴𝑅𝑥𝑥𝑅𝑥))
1712, 16mpbidi 241 . . . 4 (𝐴𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥𝑅𝑥))
187, 17jcad 512 . . 3 (𝐴𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → (𝑥 = 𝐴𝑥𝑅𝑥)))
19 brressn 38422 . . . 4 ((𝑥 ∈ V ∧ 𝑥 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴𝑥𝑅𝑥)))
2019el2v 3484 . . 3 (𝑥(𝑅 ↾ {𝐴})𝑥 ↔ (𝑥 = 𝐴𝑥𝑅𝑥))
2118, 20imbitrrdi 252 . 2 (𝐴𝑉 → (𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴})) → 𝑥(𝑅 ↾ {𝐴})𝑥))
2221ralrimiv 3142 1 (𝐴𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  cin 3961  {csn 4630   class class class wbr 5147  dom cdm 5688  ran crn 5689  cres 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700
This theorem is referenced by:  refrelressn  38505
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