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| Mirrors > Home > MPE Home > Th. List > dmsnopss | Structured version Visualization version GIF version | ||
| Description: The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on 𝐵). (Contributed by Mario Carneiro, 30-Apr-2015.) |
| Ref | Expression |
|---|---|
| dmsnopss | ⊢ dom {〈𝐴, 𝐵〉} ⊆ {𝐴} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmsnopg 6233 | . . 3 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
| 2 | eqimss 4042 | . . 3 ⊢ (dom {〈𝐴, 𝐵〉} = {𝐴} → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) |
| 4 | opprc2 4898 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) | |
| 5 | 4 | sneqd 4638 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → {〈𝐴, 𝐵〉} = {∅}) |
| 6 | 5 | dmeqd 5916 | . . . 4 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = dom {∅}) |
| 7 | dmsn0 6229 | . . . 4 ⊢ dom {∅} = ∅ | |
| 8 | 6, 7 | eqtrdi 2793 | . . 3 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = ∅) |
| 9 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ {𝐴} | |
| 10 | 8, 9 | eqsstrdi 4028 | . 2 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) |
| 11 | 3, 10 | pm2.61i 182 | 1 ⊢ dom {〈𝐴, 𝐵〉} ⊆ {𝐴} |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 {csn 4626 〈cop 4632 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 |
| This theorem is referenced by: snopsuppss 8204 strle1 17195 setsres 17215 setscom 17217 setsid 17244 ex-res 30460 bj-fununsn1 37254 mapfzcons1 42728 |
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