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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 2724 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 2724 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opth1 4208 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 → 𝑥 = 𝐴) |
4 | 3 | exlimiv 1585 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 → 𝑥 = 𝐴) |
5 | opeq1 3752 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉) | |
6 | opeq2 3753 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝐵〉) | |
7 | 6 | eqeq1d 2173 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉)) |
8 | 7 | spcegv 2809 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → (〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉 → ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉)) |
9 | 5, 8 | syl5 32 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥 = 𝐴 → ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉)) |
10 | 4, 9 | impbid2 142 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 𝑥 = 𝐴)) |
11 | 1 | eldm2 4796 | . . . 4 ⊢ (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉}) |
12 | 1, 2 | opex 4201 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V |
13 | 12 | elsn 3586 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
14 | 13 | exbii 1592 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
15 | 11, 14 | bitri 183 | . . 3 ⊢ (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
16 | velsn 3587 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
17 | 10, 15, 16 | 3bitr4g 222 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ 𝑥 ∈ {𝐴})) |
18 | 17 | eqrdv 2162 | 1 ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∃wex 1479 ∈ wcel 2135 {csn 3570 〈cop 3573 dom cdm 4598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-dm 4608 |
This theorem is referenced by: dmpropg 5070 dmsnop 5071 rnsnopg 5076 elxp4 5085 fnsng 5229 funprg 5232 funtpg 5233 fntpg 5238 ennnfonelemhdmp1 12279 ennnfonelemkh 12282 setsvala 12362 setsresg 12369 setscom 12371 setsslid 12381 strle1g 12421 |
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