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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 5283 | . . . 4 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}) | |
2 | vex 3482 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | sseq2 4022 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦)) | |
4 | treq 5273 | . . . . . . 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 3681 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)) |
7 | 6 | simprbi 496 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} → Tr 𝑦) |
8 | 1, 7 | mprg 3065 | . . 3 ⊢ Tr ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} |
9 | tcvalg 9776 | . . . 4 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}) | |
10 | treq 5273 | . . . 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 258 | . 2 ⊢ (𝐴 ∈ V → Tr (TC‘𝐴)) |
13 | tr0 5278 | . . 3 ⊢ Tr ∅ | |
14 | fvprc 6899 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∅) | |
15 | treq 5273 | . . . 4 ⊢ ((TC‘𝐴) = ∅ → (Tr (TC‘𝐴) ↔ Tr ∅)) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Tr (TC‘𝐴) ↔ Tr ∅)) |
17 | 13, 16 | mpbiri 258 | . 2 ⊢ (¬ 𝐴 ∈ V → Tr (TC‘𝐴)) |
18 | 12, 17 | pm2.61i 182 | 1 ⊢ Tr (TC‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 ∩ cint 4951 Tr wtr 5265 ‘cfv 6563 TCctc 9774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-tc 9775 |
This theorem is referenced by: tc2 9780 tcidm 9784 itunitc1 10458 |
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