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Mathbox for Thierry Arnoux |
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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 6245 | . . . 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 6245 | . . . 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 4179 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (◡{〈𝐴, 𝐵〉} ∪ ◡{〈𝐶, 𝐷〉}) = ({〈𝐵, 𝐴〉} ∪ {〈𝐷, 𝐶〉})) |
6 | df-pr 4634 | . . . 4 ⊢ {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) | |
7 | 6 | cnveqi 5888 | . . 3 ⊢ ◡{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = ◡({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) |
8 | cnvun 6165 | . . 3 ⊢ ◡({〈𝐴, 𝐵〉} ∪ {〈𝐶, 𝐷〉}) = (◡{〈𝐴, 𝐵〉} ∪ ◡{〈𝐶, 𝐷〉}) | |
9 | 7, 8 | eqtri 2763 | . 2 ⊢ ◡{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = (◡{〈𝐴, 𝐵〉} ∪ ◡{〈𝐶, 𝐷〉}) |
10 | df-pr 4634 | . 2 ⊢ {〈𝐵, 𝐴〉, 〈𝐷, 𝐶〉} = ({〈𝐵, 𝐴〉} ∪ {〈𝐷, 𝐶〉}) | |
11 | 5, 9, 10 | 3eqtr4g 2800 | 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 1537 ∈ wcel 2106 ∪ cun 3961 {csn 4631 {cpr 4633 〈cop 4637 ◡ccnv 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 |
This theorem is referenced by: cycpm2tr 33122 |
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