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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 2139 | . . . . 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 5693 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) ⊆ (𝑆 × 𝑅)) |
5 | relxp 4648 | . . . 4 ⊢ Rel (𝑆 × 𝑅) | |
6 | relss 4626 | . . . 4 ⊢ (ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉))) | |
7 | 4, 5, 6 | mpisyl 1422 | . . 3 ⊢ (𝜑 → Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
8 | relcnv 4917 | . . 3 ⊢ Rel ◡𝐹 | |
9 | 7, 8 | jctil 310 | . 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 5694 | . . . . . 6 ⊢ (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
12 | vex 2689 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
13 | vex 2689 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
14 | 12, 13 | brcnv 4722 | . . . . . 6 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦) |
15 | ancom 264 | . . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) | |
16 | 15 | rexbii 2442 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) |
17 | 11, 14, 16 | 3bitr4g 222 | . . . . 5 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴))) |
18 | 1, 2, 3 | fliftel 5694 | . . . . 5 ⊢ (𝜑 → (𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴))) |
19 | 17, 18 | bitr4d 190 | . . . 4 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ 𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧)) |
20 | df-br 3930 | . . . 4 ⊢ (𝑦◡𝐹𝑧 ↔ 〈𝑦, 𝑧〉 ∈ ◡𝐹) | |
21 | df-br 3930 | . . . 4 ⊢ (𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧 ↔ 〈𝑦, 𝑧〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) | |
22 | 19, 20, 21 | 3bitr3g 221 | . . 3 ⊢ (𝜑 → (〈𝑦, 𝑧〉 ∈ ◡𝐹 ↔ 〈𝑦, 𝑧〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉))) |
23 | 22 | eqrelrdv2 4638 | . 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) ∧ 𝜑) → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
24 | 9, 23 | mpancom 418 | 1 ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ⊆ wss 3071 〈cop 3530 class class class wbr 3929 ↦ cmpt 3989 × cxp 4537 ◡ccnv 4538 ran crn 4540 Rel wrel 4544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 |
This theorem is referenced by: (None) |
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