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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, cf. symrefref3 34940. (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 5599 | . . 3 ⊢ (◡𝑅 ⊆ 𝑅 → ran ◡𝑅 ⊆ ran 𝑅) | |
2 | rncnv 34702 | . . . . 5 ⊢ ran ◡𝑅 = dom 𝑅 | |
3 | 2 | sseq1i 3848 | . . . 4 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅) |
4 | 3 | biimpi 208 | . . 3 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅) |
5 | idreseqidinxp 34711 | . . 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 3851 | 1 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∩ cin 3791 ⊆ wss 3792 I cid 5260 × cxp 5353 ◡ccnv 5354 dom cdm 5355 ran crn 5356 ↾ cres 5357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-dm 5365 df-rn 5366 df-res 5367 |
This theorem is referenced by: symrefref3 34940 refsymrels2 34941 refsymrel2 34943 |
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