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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 5835 | . 2 ⊢ Rel ◡𝑅 | |
2 | relxp 5453 | . . . 4 ⊢ Rel ({𝑥} × (◡𝑅 “ {𝑥})) | |
3 | 2 | rgenw 3115 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥})) |
4 | reliun 5567 | . . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥}))) | |
5 | 3, 4 | mpbir 232 | . 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) |
6 | vex 3435 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
7 | vex 3435 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | opeldm 5654 | . . . . . . . 8 ⊢ (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ dom ◡𝑅) |
9 | df-rn 5446 | . . . . . . . 8 ⊢ ran 𝑅 = dom ◡𝑅 | |
10 | 8, 9 | syl6eleqr 2892 | . . . . . . 7 ⊢ (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ ran 𝑅) |
11 | ssel2 3879 | . . . . . . 7 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 𝑧 ∈ ran 𝑅) → 𝑧 ∈ 𝐴) | |
12 | 10, 11 | sylan2 592 | . . . . . 6 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅) → 𝑧 ∈ 𝐴) |
13 | 12 | ex 413 | . . . . 5 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ 𝐴)) |
14 | 13 | pm4.71rd 563 | . . . 4 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅))) |
15 | 6, 7 | elimasn 5822 | . . . . 5 ⊢ (𝑦 ∈ (◡𝑅 “ {𝑧}) ↔ 〈𝑧, 𝑦〉 ∈ ◡𝑅) |
16 | 15 | anbi2i 622 | . . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})) ↔ (𝑧 ∈ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅)) |
17 | 14, 16 | syl6bbr 290 | . . 3 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})))) |
18 | sneq 4476 | . . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧}) | |
19 | 18 | imaeq2d 5798 | . . . 4 ⊢ (𝑥 = 𝑧 → (◡𝑅 “ {𝑥}) = (◡𝑅 “ {𝑧})) |
20 | 19 | opeliunxp2 5587 | . . 3 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧}))) |
21 | 17, 20 | syl6bbr 290 | . 2 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))) |
22 | 1, 5, 21 | eqrelrdv 5543 | 1 ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ∀wral 3103 ⊆ wss 3854 {csn 4466 〈cop 4472 ∪ ciun 4819 × cxp 5433 ◡ccnv 5434 dom cdm 5435 ran crn 5436 “ cima 5438 Rel wrel 5440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pr 5214 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-iun 4821 df-br 4957 df-opab 5019 df-xp 5441 df-rel 5442 df-cnv 5443 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 |
This theorem is referenced by: gsummpt2co 30453 |
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