![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6201 | . . . . 5 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | 1 | sneqd 3620 | . . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
3 | 2 | cnveqd 4821 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ◡{𝐴} = ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
4 | 3 | unieqd 3835 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉}) |
5 | 1stexg 6193 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ V) | |
6 | 2ndexg 6194 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ V) | |
7 | opswapg 5133 | . . 3 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) → ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) | |
8 | 5, 6, 7 | syl2anc 411 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{〈(1st ‘𝐴), (2nd ‘𝐴)〉} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) |
9 | 4, 8 | eqtrd 2222 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 {csn 3607 〈cop 3610 ∪ cuni 3824 × cxp 4642 ◡ccnv 4643 ‘cfv 5235 1st c1st 6164 2nd c2nd 6165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fo 5241 df-fv 5243 df-1st 6166 df-2nd 6167 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |