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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvprop | Structured version Visualization version GIF version | ||
| Description: Converse of a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| Ref | Expression |
|---|---|
| cnvprop | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {〈𝐵, 𝐴〉, 〈𝐷, 𝐶〉}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnvsng 6243 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) |
| 3 | cnvsng 6243 | . . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ◡{〈𝐶, 𝐷〉} = {〈𝐷, 𝐶〉}) | |
| 4 | 3 | adantl 481 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{〈𝐶, 𝐷〉} = {〈𝐷, 𝐶〉}) |
| 5 | 2, 4 | uneq12d 4169 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (◡{〈𝐴, 𝐵〉} ∪ ◡{〈𝐶, 𝐷〉}) = ({〈𝐵, 𝐴〉} ∪ {〈𝐷, 𝐶〉})) |
| 6 | df-pr 4629 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) | |
| 7 | 6 | cnveqi 5885 | . . 3 ⊢ ◡{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ◡({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) |
| 8 | cnvun 6162 | . . 3 ⊢ ◡({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) = (◡{〈𝐴, 𝐵〉} ∪ ◡{〈𝐶, 𝐷〉}) | |
| 9 | 7, 8 | eqtri 2765 | . 2 ⊢ ◡{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = (◡{〈𝐴, 𝐵〉} ∪ ◡{〈𝐶, 𝐷〉}) |
| 10 | df-pr 4629 | . 2 ⊢ {〈𝐵, 𝐴〉, 〈𝐷, 𝐶〉} = ({〈𝐵, 𝐴〉} ∪ {〈𝐷, 𝐶〉}) | |
| 11 | 5, 9, 10 | 3eqtr4g 2802 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = {〈𝐵, 𝐴〉, 〈𝐷, 𝐶〉}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 {cpr 4628 〈cop 4632 ◡ccnv 5684 |
| 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-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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: cycpm2tr 33139 |
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