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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 4626 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
2 | ssel 3131 | . . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V))) | |
3 | 1, 2 | mtoi 654 | . . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹) |
4 | df-rel 4605 | . . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V)) | |
5 | reldmtpos 6212 | . . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹) | |
6 | 3, 4, 5 | 3imtr4i 200 | . . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹) |
7 | relcnv 4976 | . . 3 ⊢ Rel ◡dom 𝐹 | |
8 | 6, 7 | jctir 311 | . 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹)) |
9 | vex 2724 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | vex 2724 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
11 | vex 2724 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
12 | brtposg 6213 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ 〈𝑦, 𝑥〉𝐹𝑧)) | |
13 | 9, 10, 11, 12 | mp3an 1326 | . . . . . 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 1812 | . . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧)) |
16 | 9, 10 | opex 4201 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ∈ V |
17 | 16 | eldm 4795 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ dom tpos 𝐹 ↔ ∃𝑧〈𝑥, 𝑦〉tpos 𝐹𝑧) |
18 | 9, 10 | opelcnv 4780 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡dom 𝐹 ↔ 〈𝑦, 𝑥〉 ∈ dom 𝐹) |
19 | 10, 9 | opex 4201 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V |
20 | 19 | eldm 4795 | . . . . 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 4697 | . 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹) |
24 | 8, 23 | mpancom 419 | 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 1342 ∃wex 1479 ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 ∅c0 3404 〈cop 3573 class class class wbr 3976 × cxp 4596 ◡ccnv 4597 dom cdm 4598 Rel wrel 4603 tpos ctpos 6203 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 df-tpos 6204 |
This theorem is referenced by: rntpos 6216 dftpos2 6220 dftpos3 6221 tposfn2 6225 |
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