![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > fliftcnv | GIF version |
Description: Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
flift.1 | ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) |
flift.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) |
flift.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) |
Ref | Expression |
---|---|
fliftcnv | ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2040 | . . . . 5 ⊢ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) | |
2 | flift.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) | |
3 | flift.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) | |
4 | 1, 2, 3 | fliftrel 5432 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) ⊆ (𝑆 × 𝑅)) |
5 | relxp 4447 | . . . 4 ⊢ Rel (𝑆 × 𝑅) | |
6 | relss 4427 | . . . 4 ⊢ (ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉))) | |
7 | 4, 5, 6 | mpisyl 1335 | . . 3 ⊢ (𝜑 → Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
8 | relcnv 4703 | . . 3 ⊢ Rel ◡𝐹 | |
9 | 7, 8 | jctil 295 | . 2 ⊢ (𝜑 → (Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉))) |
10 | flift.1 | . . . . . . 7 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) | |
11 | 10, 3, 2 | fliftel 5433 | . . . . . 6 ⊢ (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
12 | vex 2560 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
13 | vex 2560 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
14 | 12, 13 | brcnv 4518 | . . . . . 6 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦) |
15 | ancom 253 | . . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) | |
16 | 15 | rexbii 2331 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) |
17 | 11, 14, 16 | 3bitr4g 212 | . . . . 5 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴))) |
18 | 1, 2, 3 | fliftel 5433 | . . . . 5 ⊢ (𝜑 → (𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴))) |
19 | 17, 18 | bitr4d 180 | . . . 4 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ 𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧)) |
20 | df-br 3765 | . . . 4 ⊢ (𝑦◡𝐹𝑧 ↔ 〈𝑦, 𝑧〉 ∈ ◡𝐹) | |
21 | df-br 3765 | . . . 4 ⊢ (𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧 ↔ 〈𝑦, 𝑧〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) | |
22 | 19, 20, 21 | 3bitr3g 211 | . . 3 ⊢ (𝜑 → (〈𝑦, 𝑧〉 ∈ ◡𝐹 ↔ 〈𝑦, 𝑧〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉))) |
23 | 22 | eqrelrdv2 4439 | . 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) ∧ 𝜑) → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
24 | 9, 23 | mpancom 399 | 1 ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∃wrex 2307 ⊆ wss 2917 〈cop 3378 class class class wbr 3764 ↦ cmpt 3818 × cxp 4343 ◡ccnv 4344 ran crn 4346 Rel wrel 4350 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |