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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 5524 | . . 3 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
2 | eqimss 3620 | . . 3 ⊢ (dom {〈𝐴, 𝐵〉} = {𝐴} → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) |
4 | opprc2 4364 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) | |
5 | 4 | sneqd 4137 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → {〈𝐴, 𝐵〉} = {∅}) |
6 | 5 | dmeqd 5248 | . . . 4 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = dom {∅}) |
7 | dmsn0 5520 | . . . 4 ⊢ dom {∅} = ∅ | |
8 | 6, 7 | syl6eq 2660 | . . 3 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = ∅) |
9 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ {𝐴} | |
10 | 8, 9 | syl6eqss 3618 | . 2 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) |
11 | 3, 10 | pm2.61i 175 | 1 ⊢ dom {〈𝐴, 𝐵〉} ⊆ {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 {csn 4125 〈cop 4131 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 |
This theorem is referenced by: snopsuppss 7197 setsres 15729 setscom 15731 setsid 15742 strlemor1 15796 strle1 15800 constr3pthlem1 26183 ex-res 26690 mapfzcons1 36298 |
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