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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 6235 | . . 3 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
2 | eqimss 4054 | . . 3 ⊢ (dom {〈𝐴, 𝐵〉} = {𝐴} → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) |
4 | opprc2 4903 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) | |
5 | 4 | sneqd 4643 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → {〈𝐴, 𝐵〉} = {∅}) |
6 | 5 | dmeqd 5919 | . . . 4 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = dom {∅}) |
7 | dmsn0 6231 | . . . 4 ⊢ dom {∅} = ∅ | |
8 | 6, 7 | eqtrdi 2791 | . . 3 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = ∅) |
9 | 0ss 4406 | . . 3 ⊢ ∅ ⊆ {𝐴} | |
10 | 8, 9 | eqsstrdi 4050 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 {csn 4631 〈cop 4637 dom cdm 5689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-dm 5699 |
This theorem is referenced by: snopsuppss 8203 strle1 17192 setsres 17212 setscom 17214 setsid 17242 ex-res 30470 bj-fununsn1 37236 mapfzcons1 42705 |
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