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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 36452. (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 5826 | . . 3 ⊢ (◡𝑅 ⊆ 𝑅 → ran ◡𝑅 ⊆ ran 𝑅) | |
2 | rncnv 36210 | . . . . 5 ⊢ ran ◡𝑅 = dom 𝑅 | |
3 | 2 | sseq1i 3946 | . . . 4 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅) |
4 | 3 | biimpi 219 | . . 3 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅) |
5 | idreseqidinxp 36219 | . . 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 3949 | 1 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∩ cin 3882 ⊆ wss 3883 I cid 5471 × cxp 5567 ◡ccnv 5568 dom cdm 5569 ran crn 5570 ↾ cres 5571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pr 5339 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-id 5472 df-xp 5575 df-rel 5576 df-cnv 5577 df-dm 5579 df-rn 5580 df-res 5581 |
This theorem is referenced by: symrefref3 36452 refsymrels2 36453 refsymrel2 36455 |
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