![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2660 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 2660 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opth1 4118 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 → 𝑥 = 𝐴) |
4 | 3 | exlimiv 1560 | . . . 4 ⊢ (∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 → 𝑥 = 𝐴) |
5 | opeq1 3671 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉) | |
6 | opeq2 3672 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝐵〉) | |
7 | 6 | eqeq1d 2123 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉)) |
8 | 7 | spcegv 2745 | . . . . 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 4697 | . . . 4 ⊢ (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉}) |
12 | 1, 2 | opex 4111 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V |
13 | 12 | elsn 3509 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
14 | 13 | exbii 1567 | . . . 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 3510 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
17 | 10, 15, 16 | 3bitr4g 222 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ 𝑥 ∈ {𝐴})) |
18 | 17 | eqrdv 2113 | 1 ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 ∃wex 1451 ∈ wcel 1463 {csn 3493 〈cop 3496 dom cdm 4499 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-br 3896 df-dm 4509 |
This theorem is referenced by: dmpropg 4969 dmsnop 4970 rnsnopg 4975 elxp4 4984 fnsng 5128 funprg 5131 funtpg 5132 fntpg 5137 ennnfonelemhdmp1 11767 ennnfonelemkh 11770 setsvala 11833 setsresg 11840 setscom 11842 setsslid 11852 strle1g 11892 |
Copyright terms: Public domain | W3C validator |