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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcnv2 | Structured version Visualization version GIF version | ||
| Description: Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.) |
| Ref | Expression |
|---|---|
| dfcnv2 | ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relcnv 6122 | . 2 ⊢ Rel ◡𝑅 | |
| 2 | relxp 5703 | . . . 4 ⊢ Rel ({𝑥} × (◡𝑅 “ {𝑥})) | |
| 3 | 2 | rgenw 3065 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥})) |
| 4 | reliun 5826 | . . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥}))) | |
| 5 | 3, 4 | mpbir 231 | . 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) |
| 6 | vex 3484 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
| 7 | vex 3484 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
| 8 | 6, 7 | opeldm 5918 | . . . . . . . 8 ⊢ (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ dom ◡𝑅) |
| 9 | df-rn 5696 | . . . . . . . 8 ⊢ ran 𝑅 = dom ◡𝑅 | |
| 10 | 8, 9 | eleqtrrdi 2852 | . . . . . . 7 ⊢ (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ ran 𝑅) |
| 11 | ssel2 3978 | . . . . . . 7 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 𝑧 ∈ ran 𝑅) → 𝑧 ∈ 𝐴) | |
| 12 | 10, 11 | sylan2 593 | . . . . . 6 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅) → 𝑧 ∈ 𝐴) |
| 13 | 12 | ex 412 | . . . . 5 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ 𝐴)) |
| 14 | 13 | pm4.71rd 562 | . . . 4 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅))) |
| 15 | 6, 7 | elimasn 6108 | . . . . 5 ⊢ (𝑦 ∈ (◡𝑅 “ {𝑧}) ↔ 〈𝑧, 𝑦〉 ∈ ◡𝑅) |
| 16 | 15 | anbi2i 623 | . . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})) ↔ (𝑧 ∈ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅)) |
| 17 | 14, 16 | bitr4di 289 | . . 3 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})))) |
| 18 | sneq 4636 | . . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧}) | |
| 19 | 18 | imaeq2d 6078 | . . . 4 ⊢ (𝑥 = 𝑧 → (◡𝑅 “ {𝑥}) = (◡𝑅 “ {𝑧})) |
| 20 | 19 | opeliunxp2 5849 | . . 3 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧}))) |
| 21 | 17, 20 | bitr4di 289 | . 2 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))) |
| 22 | 1, 5, 21 | eqrelrdv 5802 | 1 ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 {csn 4626 〈cop 4632 ∪ ciun 4991 × cxp 5683 ◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: gsummpt2co 33051 |
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