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Mirrors > Home > MPE Home > Th. List > tctr | Structured version Visualization version GIF version |
Description: Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.) |
Ref | Expression |
---|---|
tctr | ⊢ Tr (TC‘𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trint 5188 | . . . 4 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}) | |
2 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | sseq2 3993 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦)) | |
4 | treq 5178 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦)) | |
5 | 3, 4 | anbi12d 632 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦))) |
6 | 2, 5 | elab 3667 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)) |
7 | 6 | simprbi 499 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → Tr 𝑦) |
8 | 1, 7 | mprg 3152 | . . 3 ⊢ Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} |
9 | tcvalg 9180 | . . . 4 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}) | |
10 | treq 5178 | . . . 4 ⊢ ((TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → (Tr (TC‘𝐴) ↔ Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})) |
12 | 8, 11 | mpbiri 260 | . 2 ⊢ (𝐴 ∈ V → Tr (TC‘𝐴)) |
13 | tr0 5183 | . . 3 ⊢ Tr ∅ | |
14 | fvprc 6663 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∅) | |
15 | treq 5178 | . . . 4 ⊢ ((TC‘𝐴) = ∅ → (Tr (TC‘𝐴) ↔ Tr ∅)) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr ∅)) |
17 | 13, 16 | mpbiri 260 | . 2 ⊢ (¬ 𝐴 ∈ V → Tr (TC‘𝐴)) |
18 | 12, 17 | pm2.61i 184 | 1 ⊢ Tr (TC‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2799 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 ∩ cint 4876 Tr wtr 5172 ‘cfv 6355 TCctc 9178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-tc 9179 |
This theorem is referenced by: tc2 9184 tcidm 9188 itunitc1 9842 |
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