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Mirrors > Home > MPE Home > Th. List > tcidm | Structured version Visualization version GIF version |
Description: The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.) |
Ref | Expression |
---|---|
tcidm | ⊢ (TC‘(TC‘𝐴)) = (TC‘𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3947 | . . 3 ⊢ (TC‘𝐴) ⊆ (TC‘𝐴) | |
2 | tctr 9481 | . . 3 ⊢ Tr (TC‘𝐴) | |
3 | fvex 6781 | . . . 4 ⊢ (TC‘𝐴) ∈ V | |
4 | tcmin 9482 | . . . 4 ⊢ ((TC‘𝐴) ∈ V → (((TC‘𝐴) ⊆ (TC‘𝐴) ∧ Tr (TC‘𝐴)) → (TC‘(TC‘𝐴)) ⊆ (TC‘𝐴))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (((TC‘𝐴) ⊆ (TC‘𝐴) ∧ Tr (TC‘𝐴)) → (TC‘(TC‘𝐴)) ⊆ (TC‘𝐴)) |
6 | 1, 2, 5 | mp2an 688 | . 2 ⊢ (TC‘(TC‘𝐴)) ⊆ (TC‘𝐴) |
7 | tcid 9480 | . . 3 ⊢ ((TC‘𝐴) ∈ V → (TC‘𝐴) ⊆ (TC‘(TC‘𝐴))) | |
8 | 3, 7 | ax-mp 5 | . 2 ⊢ (TC‘𝐴) ⊆ (TC‘(TC‘𝐴)) |
9 | 6, 8 | eqssi 3941 | 1 ⊢ (TC‘(TC‘𝐴)) = (TC‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 Vcvv 3430 ⊆ wss 3891 Tr wtr 5195 ‘cfv 6430 TCctc 9477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7579 ax-inf2 9360 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-ov 7271 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-tc 9478 |
This theorem is referenced by: (None) |
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