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Mirrors > Home > ILE Home > Th. List > 2nd1st | 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 5945 | . . . . 5 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1 | sneqd 3459 | . . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
3 | 2 | cnveqd 4612 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ◡{𝐴} = ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
4 | 3 | unieqd 3664 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
5 | 1stexg 5938 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ V) | |
6 | 2ndexg 5939 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ V) | |
7 | opswapg 4917 | . . 3 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) → ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) | |
8 | 5, 6, 7 | syl2anc 403 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) |
9 | 4, 8 | eqtrd 2120 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ∈ wcel 1438 Vcvv 2619 {csn 3446 〈cop 3449 ∪ cuni 3653 × cxp 4436 ◡ccnv 4437 ‘cfv 5015 1st c1st 5909 2nd c2nd 5910 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-sbc 2841 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-fo 5021 df-fv 5023 df-1st 5911 df-2nd 5912 |
This theorem is referenced by: (None) |
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