Theorem symrefref2 34632

Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, cf. symrefref3 34633. (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 5509	. . 3 ⊢ (◡𝑅 ⊆ 𝑅 → ran ◡𝑅 ⊆ ran 𝑅)
2		rncnv 34394	. . . . 5 ⊢ ran ◡𝑅 = dom 𝑅
3	2	sseq1i 3770	. . . 4 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅)
4	3	biimpi 206	. . 3 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅)
5		idreseqidinxp 34404	. . 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 3773	1 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632   ∩ cin 3714   ⊆ wss 3715   I cid 5173   × cxp 5264  ^◡ccnv 5265  dom cdm 5266  ran crn 5267   ↾ cres 5268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-fun 6051  df-fn 6052

This theorem is referenced by:  symrefref3  34633  refsymrels2  34634  refsymrel2  34636