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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 7998 | . . . . 5 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1 | sneqd 4635 | . . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
3 | 2 | cnveqd 5868 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ◡{𝐴} = ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
4 | 3 | unieqd 4916 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
5 | opswap 6218 | . 2 ⊢ ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉 | |
6 | 4, 5 | eqtrdi 2788 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4623 〈cop 4629 ∪ cuni 4902 × cxp 5668 ◡ccnv 5669 ‘cfv 6533 1st c1st 7957 2nd c2nd 7958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-iota 6485 df-fun 6535 df-fv 6541 df-1st 7959 df-2nd 7960 |
This theorem is referenced by: fcnvgreu 31831 gsumhashmul 32143 |
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