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Theorem symrefref2 37054
Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 37055. (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 5899 . . 3 (𝑅𝑅 → ran 𝑅 ⊆ ran 𝑅)
2 rncnv 36790 . . . . 5 ran 𝑅 = dom 𝑅
32sseq1i 3977 . . . 4 (ran 𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅)
43biimpi 215 . . 3 (ran 𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅)
5 idreseqidinxp 36799 . . 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 3980 1 (𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  cin 3914  wss 3915   I cid 5535   × cxp 5636  ccnv 5637  dom cdm 5638  ran crn 5639  cres 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650
This theorem is referenced by:  symrefref3  37055  refsymrels2  37056  refsymrel2  37058
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