Theorem symrefref2 39672

Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 39673. (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 5920	. . 3 ⊢ (◡𝑅 ⊆ 𝑅 → ran ◡𝑅 ⊆ ran 𝑅)
2		rncnv 39331	. . . . 5 ⊢ ran ◡𝑅 = dom 𝑅
3	2	sseq1i 3936	. . . 4 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅)
4	3	biimpi 219	. . 3 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅)
5		idreseqidinxp 39340	. . 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 3939	1 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ⊆ wss 3895   ∩ cin 4089   I cid 5544   × cxp 5648  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662

This theorem is used by:  symrefref3  39673  refsymrels2  39674  refsymrel2  39676