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Mirrors > Home > MPE Home > Th. List > fliftcnv | Structured version Visualization version 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 2610 | . . . . 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 6458 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) ⊆ (𝑆 × 𝑅)) |
5 | relxp 5150 | . . . 4 ⊢ Rel (𝑆 × 𝑅) | |
6 | relss 5129 | . . . 4 ⊢ (ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉))) | |
7 | 4, 5, 6 | mpisyl 21 | . . 3 ⊢ (𝜑 → Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
8 | relcnv 5422 | . . 3 ⊢ Rel ◡𝐹 | |
9 | 7, 8 | jctil 558 | . 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 6459 | . . . . . 6 ⊢ (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) |
12 | vex 3176 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
13 | vex 3176 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
14 | 12, 13 | brcnv 5227 | . . . . . 6 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦) |
15 | ancom 465 | . . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) | |
16 | 15 | rexbii 3023 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) |
17 | 11, 14, 16 | 3bitr4g 302 | . . . . 5 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴))) |
18 | 1, 2, 3 | fliftel 6459 | . . . . 5 ⊢ (𝜑 → (𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴))) |
19 | 17, 18 | bitr4d 270 | . . . 4 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ 𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧)) |
20 | df-br 4584 | . . . 4 ⊢ (𝑦◡𝐹𝑧 ↔ 〈𝑦, 𝑧〉 ∈ ◡𝐹) | |
21 | df-br 4584 | . . . 4 ⊢ (𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧 ↔ 〈𝑦, 𝑧〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) | |
22 | 19, 20, 21 | 3bitr3g 301 | . . 3 ⊢ (𝜑 → (〈𝑦, 𝑧〉 ∈ ◡𝐹 ↔ 〈𝑦, 𝑧〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉))) |
23 | 22 | eqrelrdv2 5142 | . 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) ∧ 𝜑) → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
24 | 9, 23 | mpancom 700 | 1 ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ◡ccnv 5037 ran crn 5039 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 |
This theorem is referenced by: pi1xfrcnvlem 22664 |
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