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Mirrors > Home > MPE Home > Th. List > 2nd1st | Structured version Visualization version GIF version |
Description: Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) |
Ref | Expression |
---|---|
2nd1st | ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd2 7722 | . . . . 5 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1 | sneqd 4573 | . . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
3 | 2 | cnveqd 5741 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ◡{𝐴} = ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
4 | 3 | unieqd 4842 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
5 | opswap 6081 | . 2 ⊢ ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉 | |
6 | 4, 5 | syl6eq 2872 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {csn 4561 〈cop 4567 ∪ cuni 4832 × cxp 5548 ◡ccnv 5549 ‘cfv 6350 1st c1st 7681 2nd c2nd 7682 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fv 6358 df-1st 7683 df-2nd 7684 |
This theorem is referenced by: fcnvgreu 30412 |
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