Mathbox for Thierry Arnoux |
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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 5422 | . 2 ⊢ Rel ◡𝑅 | |
2 | relxp 5150 | . . . 4 ⊢ Rel ({𝑥} × (◡𝑅 “ {𝑥})) | |
3 | 2 | rgenw 2908 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥})) |
4 | reliun 5162 | . . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥}))) | |
5 | 3, 4 | mpbir 220 | . 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) |
6 | vex 3176 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
7 | vex 3176 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | opeldm 5250 | . . . . . . . 8 ⊢ (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ dom ◡𝑅) |
9 | df-rn 5049 | . . . . . . . 8 ⊢ ran 𝑅 = dom ◡𝑅 | |
10 | 8, 9 | syl6eleqr 2699 | . . . . . . 7 ⊢ (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ ran 𝑅) |
11 | ssel2 3563 | . . . . . . 7 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 𝑧 ∈ ran 𝑅) → 𝑧 ∈ 𝐴) | |
12 | 10, 11 | sylan2 490 | . . . . . 6 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅) → 𝑧 ∈ 𝐴) |
13 | 12 | ex 449 | . . . . 5 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ 𝐴)) |
14 | 13 | pm4.71rd 665 | . . . 4 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅))) |
15 | 6, 7 | elimasn 5409 | . . . . 5 ⊢ (𝑦 ∈ (◡𝑅 “ {𝑧}) ↔ 〈𝑧, 𝑦〉 ∈ ◡𝑅) |
16 | 15 | anbi2i 726 | . . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})) ↔ (𝑧 ∈ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅)) |
17 | 14, 16 | syl6bbr 277 | . . 3 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})))) |
18 | sneq 4135 | . . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧}) | |
19 | 18 | imaeq2d 5385 | . . . 4 ⊢ (𝑥 = 𝑧 → (◡𝑅 “ {𝑥}) = (◡𝑅 “ {𝑧})) |
20 | 19 | opeliunxp2 5182 | . . 3 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧}))) |
21 | 17, 20 | syl6bbr 277 | . 2 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))) |
22 | 1, 5, 21 | eqrelrdv 5139 | 1 ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 {csn 4125 〈cop 4131 ∪ ciun 4455 × cxp 5036 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-iun 4457 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: gsummpt2co 29111 |
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