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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 5644 | . 2 ⊢ Rel ◡𝑅 | |
2 | relxp 5266 | . . . 4 ⊢ Rel ({𝑥} × (◡𝑅 “ {𝑥})) | |
3 | 2 | rgenw 3073 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥})) |
4 | reliun 5378 | . . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥}))) | |
5 | 3, 4 | mpbir 221 | . 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) |
6 | vex 3354 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
7 | vex 3354 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
8 | 6, 7 | opeldm 5466 | . . . . . . . 8 ⊢ (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ dom ◡𝑅) |
9 | df-rn 5260 | . . . . . . . 8 ⊢ ran 𝑅 = dom ◡𝑅 | |
10 | 8, 9 | syl6eleqr 2861 | . . . . . . 7 ⊢ (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ ran 𝑅) |
11 | ssel2 3747 | . . . . . . 7 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 𝑧 ∈ ran 𝑅) → 𝑧 ∈ 𝐴) | |
12 | 10, 11 | sylan2 572 | . . . . . 6 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅) → 𝑧 ∈ 𝐴) |
13 | 12 | ex 397 | . . . . 5 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 → 𝑧 ∈ 𝐴)) |
14 | 13 | pm4.71rd 544 | . . . 4 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅))) |
15 | 6, 7 | elimasn 5631 | . . . . 5 ⊢ (𝑦 ∈ (◡𝑅 “ {𝑧}) ↔ 〈𝑧, 𝑦〉 ∈ ◡𝑅) |
16 | 15 | anbi2i 601 | . . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})) ↔ (𝑧 ∈ 𝐴 ∧ 〈𝑧, 𝑦〉 ∈ ◡𝑅)) |
17 | 14, 16 | syl6bbr 278 | . . 3 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})))) |
18 | sneq 4326 | . . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧}) | |
19 | 18 | imaeq2d 5607 | . . . 4 ⊢ (𝑥 = 𝑧 → (◡𝑅 “ {𝑥}) = (◡𝑅 “ {𝑧})) |
20 | 19 | opeliunxp2 5399 | . . 3 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧}))) |
21 | 17, 20 | syl6bbr 278 | . 2 ⊢ (ran 𝑅 ⊆ 𝐴 → (〈𝑧, 𝑦〉 ∈ ◡𝑅 ↔ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))) |
22 | 1, 5, 21 | eqrelrdv 5356 | 1 ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ⊆ wss 3723 {csn 4316 〈cop 4322 ∪ ciun 4654 × cxp 5247 ◡ccnv 5248 dom cdm 5249 ran crn 5250 “ cima 5252 Rel wrel 5254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-iun 4656 df-br 4787 df-opab 4847 df-xp 5255 df-rel 5256 df-cnv 5257 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 |
This theorem is referenced by: gsummpt2co 30120 |
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