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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 5710 | . . . . 5 ⊢ ¬ ∅ ∈ (V × V) | |
2 | ssel 3975 | . . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V))) | |
3 | 1, 2 | mtoi 198 | . . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹) |
4 | df-rel 5683 | . . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V)) | |
5 | reldmtpos 8218 | . . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹) | |
6 | 3, 4, 5 | 3imtr4i 291 | . . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹) |
7 | relcnv 6103 | . . 3 ⊢ Rel ◡dom 𝐹 | |
8 | 6, 7 | jctir 521 | . 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹)) |
9 | vex 3478 | . . . . . 6 ⊢ 𝑧 ∈ V | |
10 | brtpos 8219 | . . . . . 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 1924 | . . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧)) |
13 | opex 5464 | . . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V | |
14 | 13 | eldm 5900 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧) |
15 | vex 3478 | . . . . . 6 ⊢ 𝑥 ∈ V | |
16 | vex 3478 | . . . . . 6 ⊢ 𝑦 ∈ V | |
17 | 15, 16 | opelcnv 5881 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom 𝐹) |
18 | opex 5464 | . . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V | |
19 | 18 | eldm 5900 | . . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧) |
20 | 17, 19 | bitri 274 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧) |
21 | 12, 14, 20 | 3bitr4g 313 | . . 3 ⊢ (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹)) |
22 | 21 | eqrelrdv2 5795 | . 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹) |
23 | 8, 22 | mpancom 686 | 1 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 ∅c0 4322 ⟨cop 4634 class class class wbr 5148 × cxp 5674 ◡ccnv 5675 dom cdm 5676 Rel wrel 5681 tpos ctpos 8209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-tpos 8210 |
This theorem is referenced by: rntpos 8223 dftpos2 8227 dftpos3 8228 tposfn2 8232 |
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