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| 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 2736 | . . . . 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 7329 | . . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉) ⊆ (𝑆 × 𝑅)) | 
| 5 | relxp 5702 | . . . 4 ⊢ Rel (𝑆 × 𝑅) | |
| 6 | relss 5790 | . . . 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 6121 | . . 3 ⊢ Rel ◡𝐹 | |
| 9 | 7, 8 | jctil 519 | . 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 7330 | . . . . . 6 ⊢ (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))) | 
| 12 | vex 3483 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 13 | vex 3483 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
| 14 | 12, 13 | brcnv 5892 | . . . . . 6 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦) | 
| 15 | ancom 460 | . . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) | |
| 16 | 15 | rexbii 3093 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)) | 
| 17 | 11, 14, 16 | 3bitr4g 314 | . . . . 5 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴))) | 
| 18 | 1, 2, 3 | fliftel 7330 | . . . . 5 ⊢ (𝜑 → (𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴))) | 
| 19 | 17, 18 | bitr4d 282 | . . . 4 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ 𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧)) | 
| 20 | df-br 5143 | . . . 4 ⊢ (𝑦◡𝐹𝑧 ↔ 〈𝑦, 𝑧〉 ∈ ◡𝐹) | |
| 21 | df-br 5143 | . . . 4 ⊢ (𝑦ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)𝑧 ↔ 〈𝑦, 𝑧〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) | |
| 22 | 19, 20, 21 | 3bitr3g 313 | . . 3 ⊢ (𝜑 → (〈𝑦, 𝑧〉 ∈ ◡𝐹 ↔ 〈𝑦, 𝑧〉 ∈ ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉))) | 
| 23 | 22 | eqrelrdv2 5804 | . 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) ∧ 𝜑) → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) | 
| 24 | 9, 23 | mpancom 688 | 1 ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈𝐵, 𝐴〉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ⊆ wss 3950 〈cop 4631 class class class wbr 5142 ↦ cmpt 5224 × cxp 5682 ◡ccnv 5683 ran crn 5685 Rel wrel 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: pi1xfrcnvlem 25090 | 
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