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Theorem symrefref2 38545
Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 38546. (Contributed by Peter Mazsa, 19-Jul-2018.)
Assertion
Ref Expression
symrefref2 (𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))

Proof of Theorem symrefref2
StepHypRef Expression
1 rnss 5953 . . 3 (𝑅𝑅 → ran 𝑅 ⊆ ran 𝑅)
2 rncnv 38282 . . . . 5 ran 𝑅 = dom 𝑅
32sseq1i 4024 . . . 4 (ran 𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅)
43biimpi 216 . . 3 (ran 𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅)
5 idreseqidinxp 38291 . . 3 (dom 𝑅 ⊆ ran 𝑅 → ( I ∩ (dom 𝑅 × ran 𝑅)) = ( I ↾ dom 𝑅))
61, 4, 53syl 18 . 2 (𝑅𝑅 → ( I ∩ (dom 𝑅 × ran 𝑅)) = ( I ↾ dom 𝑅))
76sseq1d 4027 1 (𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  cin 3962  wss 3963   I cid 5582   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701
This theorem is referenced by:  symrefref3  38546  refsymrels2  38547  refsymrel2  38549
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