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Mirrors > Home > ILE Home > Th. List > dmsnopg | GIF version |
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
dmsnopg | ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 2560 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opth1 3973 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 → 𝑥 = 𝐴) |
4 | 3 | exlimiv 1489 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 → 𝑥 = 𝐴) |
5 | opeq1 3549 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉) | |
6 | opeq2 3550 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝐵〉) | |
7 | 6 | eqeq1d 2048 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉)) |
8 | 7 | spcegv 2641 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → (〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉 → ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉)) |
9 | 5, 8 | syl5 28 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥 = 𝐴 → ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉)) |
10 | 4, 9 | impbid2 131 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 𝑥 = 𝐴)) |
11 | 1 | eldm2 4533 | . . . 4 ⊢ (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉}) |
12 | 1, 2 | opex 3966 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V |
13 | 12 | elsn 3391 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
14 | 13 | exbii 1496 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
15 | 11, 14 | bitri 173 | . . 3 ⊢ (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
16 | velsn 3392 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
17 | 10, 15, 16 | 3bitr4g 212 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ 𝑥 ∈ {𝐴})) |
18 | 17 | eqrdv 2038 | 1 ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∃wex 1381 ∈ wcel 1393 {csn 3375 〈cop 3378 dom cdm 4345 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-dm 4355 |
This theorem is referenced by: dmpropg 4793 dmsnop 4794 rnsnopg 4799 elxp4 4808 fnsng 4947 funprg 4949 funtpg 4950 fntpg 4955 |
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