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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 2081 | . . . . 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 5452 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) ⊆ (𝑆 × 𝑅)) |
5 | relxp 4465 | . . . 4 ⊢ Rel (𝑆 × 𝑅) | |
6 | relss 4445 | . . . 4 ⊢ (ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉))) | |
7 | 4, 5, 6 | mpisyl 1375 | . . 3 ⊢ (𝜑 → Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
8 | relcnv 4723 | . . 3 ⊢ Rel ◡𝐹 | |
9 | 7, 8 | jctil 305 | . 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 5453 | . . . . . 6 ⊢ (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
12 | vex 2604 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
13 | vex 2604 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
14 | 12, 13 | brcnv 4536 | . . . . . 6 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦) |
15 | ancom 262 | . . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) | |
16 | 15 | rexbii 2373 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) |
17 | 11, 14, 16 | 3bitr4g 221 | . . . . 5 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴))) |
18 | 1, 2, 3 | fliftel 5453 | . . . . 5 ⊢ (𝜑 → (𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴))) |
19 | 17, 18 | bitr4d 189 | . . . 4 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ 𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧)) |
20 | df-br 3786 | . . . 4 ⊢ (𝑦◡𝐹𝑧 ↔ 〈𝑦, 𝑧〉 ∈ ◡𝐹) | |
21 | df-br 3786 | . . . 4 ⊢ (𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧 ↔ 〈𝑦, 𝑧〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) | |
22 | 19, 20, 21 | 3bitr3g 220 | . . 3 ⊢ (𝜑 → (〈𝑦, 𝑧〉 ∈ ◡𝐹 ↔ 〈𝑦, 𝑧〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉))) |
23 | 22 | eqrelrdv2 4457 | . 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) ∧ 𝜑) → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
24 | 9, 23 | mpancom 413 | 1 ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 ⊆ wss 2973 〈cop 3401 class class class wbr 3785 ↦ cmpt 3839 × cxp 4361 ◡ccnv 4362 ran crn 4364 Rel wrel 4368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 |
This theorem is referenced by: (None) |
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