Theorem symrefref2 37923

Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 37924. (Contributed by Peter Mazsa, 19-Jul-2018.)

Assertion

Ref	Expression
symrefref2	⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))

Proof of Theorem symrefref2

Step	Hyp	Ref	Expression
1		rnss 5928	. . 3 ⊢ (◡𝑅 ⊆ 𝑅 → ran ◡𝑅 ⊆ ran 𝑅)
2		rncnv 37659	. . . . 5 ⊢ ran ◡𝑅 = dom 𝑅
3	2	sseq1i 4002	. . . 4 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅)
4	3	biimpi 215	. . 3 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅)
5		idreseqidinxp 37668	. . 3 ⊢ (dom 𝑅 ⊆ ran 𝑅 → ( I ∩ (dom 𝑅 × ran 𝑅)) = ( I ↾ dom 𝑅))
6	1, 4, 5	3syl 18	. 2 ⊢ (◡𝑅 ⊆ 𝑅 → ( I ∩ (dom 𝑅 × ran 𝑅)) = ( I ↾ dom 𝑅))
7	6	sseq1d 4005	1 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∩ cin 3939   ⊆ wss 3940   I cid 5563   × cxp 5664  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   ↾ cres 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678

This theorem is referenced by:  symrefref3  37924  refsymrels2  37925  refsymrel2  37927