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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 5671 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
2 | ssel 3941 | . . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V))) | |
3 | 1, 2 | mtoi 198 | . . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹) |
4 | df-rel 5644 | . . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V)) | |
5 | reldmtpos 8169 | . . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹) | |
6 | 3, 4, 5 | 3imtr4i 292 | . . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹) |
7 | relcnv 6060 | . . 3 ⊢ Rel ◡dom 𝐹 | |
8 | 6, 7 | jctir 522 | . 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹)) |
9 | vex 3451 | . . . . . 6 ⊢ 𝑧 ∈ V | |
10 | brtpos 8170 | . . . . . 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 1925 | . . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧)) |
13 | opex 5425 | . . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V | |
14 | 13 | eldm 5860 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧) |
15 | vex 3451 | . . . . . 6 ⊢ 𝑥 ∈ V | |
16 | vex 3451 | . . . . . 6 ⊢ 𝑦 ∈ V | |
17 | 15, 16 | opelcnv 5841 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom 𝐹) |
18 | opex 5425 | . . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V | |
19 | 18 | eldm 5860 | . . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧) |
20 | 17, 19 | bitri 275 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧) |
21 | 12, 14, 20 | 3bitr4g 314 | . . 3 ⊢ (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹)) |
22 | 21 | eqrelrdv2 5755 | . 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹) |
23 | 8, 22 | mpancom 687 | 1 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3447 ⊆ wss 3914 ∅c0 4286 ⟨cop 4596 class class class wbr 5109 × cxp 5635 ◡ccnv 5636 dom cdm 5637 Rel wrel 5642 tpos ctpos 8160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-fv 6508 df-tpos 8161 |
This theorem is referenced by: rntpos 8174 dftpos2 8178 dftpos3 8179 tposfn2 8183 |
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