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Mirrors > Home > MPE Home > Th. List > Mathboxes > symrefref2 | Structured version Visualization version GIF version |
Description: Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 35794. (Contributed by Peter Mazsa, 19-Jul-2018.) |
Ref | Expression |
---|---|
symrefref2 | ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnss 5803 | . . 3 ⊢ (◡𝑅 ⊆ 𝑅 → ran ◡𝑅 ⊆ ran 𝑅) | |
2 | rncnv 35552 | . . . . 5 ⊢ ran ◡𝑅 = dom 𝑅 | |
3 | 2 | sseq1i 3994 | . . . 4 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅) |
4 | 3 | biimpi 218 | . . 3 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅) |
5 | idreseqidinxp 35561 | . . 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 3997 | 1 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∩ cin 3934 ⊆ wss 3935 I cid 5453 × cxp 5547 ◡ccnv 5548 dom cdm 5549 ran crn 5550 ↾ cres 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 |
This theorem is referenced by: symrefref3 35794 refsymrels2 35795 refsymrel2 35797 |
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