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Theorem symrefref3 36334
Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 36333. (Contributed by Peter Mazsa, 23-Aug-2021.) (Proof modification is discouraged.)
Assertion
Ref Expression
symrefref3 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem symrefref3
StepHypRef Expression
1 symrefref2 36333 . 2 (𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
2 cnvsym 5958 . 2 (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
3 idinxpss 36104 . . 3 (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦))
4 idrefALT 5957 . . 3 (( I ↾ dom 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥)
53, 4bibi12i 343 . 2 ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅) ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
61, 2, 53imtr3i 294 1 (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1540   = wceq 1542  wral 3054  cin 3852  wss 3853   class class class wbr 5040   I cid 5438   × cxp 5533  ccnv 5534  dom cdm 5535  ran crn 5536  cres 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-dm 5545  df-rn 5546  df-res 5547
This theorem is referenced by:  refsymrel3  36338
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