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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 37924. (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 5928 | . . 3 ⊢ (◡𝑅 ⊆ 𝑅 → ran ◡𝑅 ⊆ ran 𝑅) | |
2 | rncnv 37659 | . . . . 5 ⊢ ran ◡𝑅 = dom 𝑅 | |
3 | 2 | sseq1i 4002 | . . . 4 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 ↔ dom 𝑅 ⊆ ran 𝑅) |
4 | 3 | biimpi 215 | . . 3 ⊢ (ran ◡𝑅 ⊆ ran 𝑅 → dom 𝑅 ⊆ ran 𝑅) |
5 | idreseqidinxp 37668 | . . 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 4005 | 1 ⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∩ cin 3939 ⊆ wss 3940 I cid 5563 × cxp 5664 ◡ccnv 5665 dom cdm 5666 ran crn 5667 ↾ cres 5668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-dm 5676 df-rn 5677 df-res 5678 |
This theorem is referenced by: symrefref3 37924 refsymrels2 37925 refsymrel2 37927 |
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