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Mirrors > Home > MPE Home > Th. List > dmtpos | Structured version Visualization version 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 5389 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
2 | ssel 3814 | . . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V))) | |
3 | 1, 2 | mtoi 191 | . . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹) |
4 | df-rel 5362 | . . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V)) | |
5 | reldmtpos 7642 | . . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹) | |
6 | 3, 4, 5 | 3imtr4i 284 | . . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹) |
7 | relcnv 5757 | . . 3 ⊢ Rel ◡dom 𝐹 | |
8 | 6, 7 | jctir 516 | . 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹)) |
9 | vex 3400 | . . . . . 6 ⊢ 𝑧 ∈ V | |
10 | brtpos 7643 | . . . . . 6 ⊢ (𝑧 ∈ V → (〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ 〈𝑦, 𝑥〉𝐹𝑧)) | |
11 | 9, 10 | mp1i 13 | . . . . 5 ⊢ (Rel dom 𝐹 → (〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ 〈𝑦, 𝑥〉𝐹𝑧)) |
12 | 11 | exbidv 1964 | . . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧〈𝑥, 𝑦〉tpos 𝐹𝑧 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧)) |
13 | opex 5164 | . . . . 5 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
14 | 13 | eldm 5566 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ dom tpos 𝐹 ↔ ∃𝑧〈𝑥, 𝑦〉tpos 𝐹𝑧) |
15 | vex 3400 | . . . . . 6 ⊢ 𝑥 ∈ V | |
16 | vex 3400 | . . . . . 6 ⊢ 𝑦 ∈ V | |
17 | 15, 16 | opelcnv 5549 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ◡dom 𝐹 ↔ 〈𝑦, 𝑥〉 ∈ dom 𝐹) |
18 | opex 5164 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V | |
19 | 18 | eldm 5566 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ dom 𝐹 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧) |
20 | 17, 19 | bitri 267 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡dom 𝐹 ↔ ∃𝑧〈𝑦, 𝑥〉𝐹𝑧) |
21 | 12, 14, 20 | 3bitr4g 306 | . . 3 ⊢ (Rel dom 𝐹 → (〈𝑥, 𝑦〉 ∈ dom tpos 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ ◡dom 𝐹)) |
22 | 21 | eqrelrdv2 5466 | . 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹) |
23 | 8, 22 | mpancom 678 | 1 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∃wex 1823 ∈ wcel 2106 Vcvv 3397 ⊆ wss 3791 ∅c0 4140 〈cop 4403 class class class wbr 4886 × cxp 5353 ◡ccnv 5354 dom cdm 5355 Rel wrel 5360 tpos ctpos 7633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-fv 6143 df-tpos 7634 |
This theorem is referenced by: rntpos 7647 dftpos2 7651 dftpos3 7652 tposfn2 7656 |
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