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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 6201 | . . 3 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
2 | eqimss 4036 | . . 3 ⊢ (dom {〈𝐴, 𝐵〉} = {𝐴} → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) |
4 | opprc2 4891 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) | |
5 | 4 | sneqd 4634 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → {〈𝐴, 𝐵〉} = {∅}) |
6 | 5 | dmeqd 5897 | . . . 4 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = dom {∅}) |
7 | dmsn0 6197 | . . . 4 ⊢ dom {∅} = ∅ | |
8 | 6, 7 | eqtrdi 2787 | . . 3 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = ∅) |
9 | 0ss 4392 | . . 3 ⊢ ∅ ⊆ {𝐴} | |
10 | 8, 9 | eqsstrdi 4032 | . 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 1541 ∈ wcel 2106 Vcvv 3473 ⊆ wss 3944 ∅c0 4318 {csn 4622 〈cop 4628 dom cdm 5669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-dm 5679 |
This theorem is referenced by: snopsuppss 8146 strle1 17073 setsres 17093 setscom 17095 setsid 17123 ex-res 29559 bj-fununsn1 35938 mapfzcons1 41226 |
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