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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 4672 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
2 | ssel 3164 | . . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V))) | |
3 | 1, 2 | mtoi 665 | . . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹) |
4 | df-rel 4651 | . . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V)) | |
5 | reldmtpos 6278 | . . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹) | |
6 | 3, 4, 5 | 3imtr4i 201 | . . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹) |
7 | relcnv 5024 | . . 3 ⊢ Rel ◡dom 𝐹 | |
8 | 6, 7 | jctir 313 | . 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹)) |
9 | vex 2755 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | vex 2755 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
11 | vex 2755 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
12 | brtposg 6279 | . . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → (〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ 〈𝑦, 𝑥〉𝐹𝑧)) | |
13 | 9, 10, 11, 12 | mp3an 1348 | . . . . . 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 1836 | . . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧)) |
16 | 9, 10 | opex 4247 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ∈ V |
17 | 16 | eldm 4842 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ dom tpos 𝐹 ↔ ∃𝑧〈𝑥, 𝑦〉tpos 𝐹𝑧) |
18 | 9, 10 | opelcnv 4827 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡dom 𝐹 ↔ 〈𝑦, 𝑥〉 ∈ dom 𝐹) |
19 | 10, 9 | opex 4247 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V |
20 | 19 | eldm 4842 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ dom 𝐹 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧) |
21 | 18, 20 | bitri 184 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡dom 𝐹 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧) |
22 | 15, 17, 21 | 3bitr4g 223 | . . 3 ⊢ (Rel dom 𝐹 → (〈𝑥, 𝑦〉 ∈ dom tpos 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ ◡dom 𝐹)) |
23 | 22 | eqrelrdv2 4743 | . 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹) |
24 | 8, 23 | mpancom 422 | 1 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1503 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 ∅c0 3437 〈cop 3610 class class class wbr 4018 × cxp 4642 ◡ccnv 4643 dom cdm 4644 Rel wrel 4649 tpos ctpos 6269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-fv 5243 df-tpos 6270 |
This theorem is referenced by: rntpos 6282 dftpos2 6286 dftpos3 6287 tposfn2 6291 |
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