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 5972 | . 2 ⊢ Rel ◡𝑅 | |
2 | relxp 5569 | . . . 4 ⊢ Rel ({𝑥} × (◡𝑅 “ {𝑥})) | |
3 | 2 | rgenw 3073 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥})) |
4 | reliun 5686 | . . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥}))) | |
5 | 3, 4 | mpbir 234 | . 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) |
6 | vex 3412 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
7 | vex 3412 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | opeldm 5776 | . . . . . . . 8 ⊢ (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ dom ◡𝑅) |
9 | df-rn 5562 | . . . . . . . 8 ⊢ ran 𝑅 = dom ◡𝑅 | |
10 | 8, 9 | eleqtrrdi 2849 | . . . . . . 7 ⊢ (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ ran 𝑅) |
11 | ssel2 3895 | . . . . . . 7 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 𝑧 ∈ ran 𝑅) → 𝑧 ∈ 𝐴) | |
12 | 10, 11 | sylan2 596 | . . . . . 6 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅) → 𝑧 ∈ 𝐴) |
13 | 12 | ex 416 | . . . . 5 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ 𝐴)) |
14 | 13 | pm4.71rd 566 | . . . 4 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅))) |
15 | 6, 7 | elimasn 5957 | . . . . 5 ⊢ (𝑦 ∈ (◡𝑅 “ {𝑧}) ↔ 〈𝑧, 𝑦〉 ∈ ◡𝑅) |
16 | 15 | anbi2i 626 | . . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})) ↔ (𝑧 ∈ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅)) |
17 | 14, 16 | bitr4di 292 | . . 3 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})))) |
18 | sneq 4551 | . . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧}) | |
19 | 18 | imaeq2d 5929 | . . . 4 ⊢ (𝑥 = 𝑧 → (◡𝑅 “ {𝑥}) = (◡𝑅 “ {𝑧})) |
20 | 19 | opeliunxp2 5707 | . . 3 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧}))) |
21 | 17, 20 | bitr4di 292 | . 2 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))) |
22 | 1, 5, 21 | eqrelrdv 5662 | 1 ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ⊆ wss 3866 {csn 4541 〈cop 4547 ∪ ciun 4904 × cxp 5549 ◡ccnv 5550 dom cdm 5551 ran crn 5552 “ cima 5554 Rel wrel 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-iun 4906 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 |
This theorem is referenced by: gsummpt2co 31027 |
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