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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 4527 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
2 | ssel 3057 | . . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V))) | |
3 | 1, 2 | mtoi 636 | . . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹) |
4 | df-rel 4506 | . . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V)) | |
5 | reldmtpos 6104 | . . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹) | |
6 | 3, 4, 5 | 3imtr4i 200 | . . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹) |
7 | relcnv 4875 | . . 3 ⊢ Rel ◡dom 𝐹 | |
8 | 6, 7 | jctir 309 | . 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹)) |
9 | vex 2660 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | vex 2660 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
11 | vex 2660 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
12 | brtposg 6105 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ 〈𝑦, 𝑥〉𝐹𝑧)) | |
13 | 9, 10, 11, 12 | mp3an 1298 | . . . . . 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 1779 | . . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧)) |
16 | 9, 10 | opex 4111 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ∈ V |
17 | 16 | eldm 4696 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ dom tpos 𝐹 ↔ ∃𝑧〈𝑥, 𝑦〉tpos 𝐹𝑧) |
18 | 9, 10 | opelcnv 4681 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡dom 𝐹 ↔ 〈𝑦, 𝑥〉 ∈ dom 𝐹) |
19 | 10, 9 | opex 4111 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V |
20 | 19 | eldm 4696 | . . . . 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 4598 | . 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹) |
24 | 8, 23 | mpancom 416 | 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 1314 ∃wex 1451 ∈ wcel 1463 Vcvv 2657 ⊆ wss 3037 ∅c0 3329 〈cop 3496 class class class wbr 3895 × cxp 4497 ◡ccnv 4498 dom cdm 4499 Rel wrel 4504 tpos ctpos 6095 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-fv 5089 df-tpos 6096 |
This theorem is referenced by: rntpos 6108 dftpos2 6112 dftpos3 6113 tposfn2 6117 |
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