Theorem symrefref2 39337

Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 39338. (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 5934	. . 3 ⊢ (◡𝑅 ⊆ 𝑅 → ran ◡𝑅 ⊆ ran 𝑅)
2		rncnv 38996	. . . . 5 ⊢ ran ◡𝑅 = dom 𝑅
3	2	sseq1i 3968	. . . 4 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅)
4	3	biimpi 219	. . 3 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅)
5		idreseqidinxp 39005	. . 3 ⊢ (dom 𝑅 ⊆ ran 𝑅 → ( I ∩ (dom 𝑅 × ran 𝑅)) = ( I ↾ dom 𝑅))
6	1, 4, 5	3syl 19	. 2 ⊢ (◡𝑅 ⊆ 𝑅 → ( I ∩ (dom 𝑅 × ran 𝑅)) = ( I ↾ dom 𝑅))
7	6	sseq1d 3971	1 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∩ cin 3907   ⊆ wss 3908   I cid 5560   × cxp 5664  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   ↾ cres 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678

This theorem is used by:  symrefref3  39338  refsymrels2  39339  refsymrel2  39341