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Mirrors > Home > ILE Home > Th. List > dmtpos | GIF version |
Description: The domain of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
dmtpos | ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 4567 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
2 | ssel 3091 | . . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V))) | |
3 | 1, 2 | mtoi 653 | . . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹) |
4 | df-rel 4546 | . . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V)) | |
5 | reldmtpos 6150 | . . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹) | |
6 | 3, 4, 5 | 3imtr4i 200 | . . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹) |
7 | relcnv 4917 | . . 3 ⊢ Rel ◡dom 𝐹 | |
8 | 6, 7 | jctir 311 | . 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹)) |
9 | vex 2689 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | vex 2689 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
11 | vex 2689 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
12 | brtposg 6151 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ 〈𝑦, 𝑥〉𝐹𝑧)) | |
13 | 9, 10, 11, 12 | mp3an 1315 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ 〈𝑦, 𝑥〉𝐹𝑧) |
14 | 13 | a1i 9 | . . . . 5 ⊢ (Rel dom 𝐹 → (〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ 〈𝑦, 𝑥〉𝐹𝑧)) |
15 | 14 | exbidv 1797 | . . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧)) |
16 | 9, 10 | opex 4151 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ∈ V |
17 | 16 | eldm 4736 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ dom tpos 𝐹 ↔ ∃𝑧〈𝑥, 𝑦〉tpos 𝐹𝑧) |
18 | 9, 10 | opelcnv 4721 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡dom 𝐹 ↔ 〈𝑦, 𝑥〉 ∈ dom 𝐹) |
19 | 10, 9 | opex 4151 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V |
20 | 19 | eldm 4736 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ dom 𝐹 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧) |
21 | 18, 20 | bitri 183 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡dom 𝐹 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧) |
22 | 15, 17, 21 | 3bitr4g 222 | . . 3 ⊢ (Rel dom 𝐹 → (〈𝑥, 𝑦〉 ∈ dom tpos 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ ◡dom 𝐹)) |
23 | 22 | eqrelrdv2 4638 | . 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹) |
24 | 8, 23 | mpancom 418 | 1 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 ∅c0 3363 〈cop 3530 class class class wbr 3929 × cxp 4537 ◡ccnv 4538 dom cdm 4539 Rel wrel 4544 tpos ctpos 6141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-tpos 6142 |
This theorem is referenced by: rntpos 6154 dftpos2 6158 dftpos3 6159 tposfn2 6163 |
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