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Mirrors > Home > MPE Home > Th. List > cottrcl | Structured version Visualization version GIF version |
Description: Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.) |
Ref | Expression |
---|---|
cottrcl | ⊢ (𝑅 ∘ t++𝑅) ⊆ t++𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 6010 | . . . . 5 ⊢ Rel (𝑅 ↾ V) | |
2 | ssttrcl 9709 | . . . . 5 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V) |
4 | coss1 5855 | . . . 4 ⊢ ((𝑅 ↾ V) ⊆ t++(𝑅 ↾ V) → ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) ⊆ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) ⊆ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) |
6 | ttrcltr 9710 | . . 3 ⊢ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) ⊆ t++(𝑅 ↾ V) | |
7 | 5, 6 | sstri 3991 | . 2 ⊢ ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) ⊆ t++(𝑅 ↾ V) |
8 | ssv 4006 | . . . 4 ⊢ ran t++(𝑅 ↾ V) ⊆ V | |
9 | cores 6248 | . . . 4 ⊢ (ran t++(𝑅 ↾ V) ⊆ V → ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) = (𝑅 ∘ t++(𝑅 ↾ V))) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) = (𝑅 ∘ t++(𝑅 ↾ V)) |
11 | ttrclresv 9711 | . . . 4 ⊢ t++(𝑅 ↾ V) = t++𝑅 | |
12 | 11 | coeq2i 5860 | . . 3 ⊢ (𝑅 ∘ t++(𝑅 ↾ V)) = (𝑅 ∘ t++𝑅) |
13 | 10, 12 | eqtri 2760 | . 2 ⊢ ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) = (𝑅 ∘ t++𝑅) |
14 | 7, 13, 11 | 3sstr3i 4024 | 1 ⊢ (𝑅 ∘ t++𝑅) ⊆ t++𝑅 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 Vcvv 3474 ⊆ wss 3948 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 Rel wrel 5681 t++cttrcl 9701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-ttrcl 9702 |
This theorem is referenced by: (None) |
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