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Theorem symrefref2 35904
Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 35905. (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 5796 . . 3 (𝑅𝑅 → ran 𝑅 ⊆ ran 𝑅)
2 rncnv 35663 . . . . 5 ran 𝑅 = dom 𝑅
32sseq1i 3981 . . . 4 (ran 𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅)
43biimpi 219 . . 3 (ran 𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅)
5 idreseqidinxp 35672 . . 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 3984 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 1538  cin 3918  wss 3919   I cid 5446   × cxp 5540  ccnv 5541  dom cdm 5542  ran crn 5543  cres 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554
This theorem is referenced by:  symrefref3  35905  refsymrels2  35906  refsymrel2  35908
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