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Theorem symrefref2 36451
Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 36452. (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 5826 . . 3 (𝑅𝑅 → ran 𝑅 ⊆ ran 𝑅)
2 rncnv 36210 . . . . 5 ran 𝑅 = dom 𝑅
32sseq1i 3946 . . . 4 (ran 𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅)
43biimpi 219 . . 3 (ran 𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅)
5 idreseqidinxp 36219 . . 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 3949 1 (𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  cin 3882  wss 3883   I cid 5471   × cxp 5567  ccnv 5568  dom cdm 5569  ran crn 5570  cres 5571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710  ax-sep 5209  ax-nul 5216  ax-pr 5339
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5472  df-xp 5575  df-rel 5576  df-cnv 5577  df-dm 5579  df-rn 5580  df-res 5581
This theorem is referenced by:  symrefref3  36452  refsymrels2  36453  refsymrel2  36455
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