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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 6072 | . . 3 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
2 | eqimss 4025 | . . 3 ⊢ (dom {〈𝐴, 𝐵〉} = {𝐴} → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐵 ∈ V → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) |
4 | opprc2 4830 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) | |
5 | 4 | sneqd 4581 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → {〈𝐴, 𝐵〉} = {∅}) |
6 | 5 | dmeqd 5776 | . . . 4 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = dom {∅}) |
7 | dmsn0 6068 | . . . 4 ⊢ dom {∅} = ∅ | |
8 | 6, 7 | syl6eq 2874 | . . 3 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} = ∅) |
9 | 0ss 4352 | . . 3 ⊢ ∅ ⊆ {𝐴} | |
10 | 8, 9 | eqsstrdi 4023 | . 2 ⊢ (¬ 𝐵 ∈ V → dom {〈𝐴, 𝐵〉} ⊆ {𝐴}) |
11 | 3, 10 | pm2.61i 184 | 1 ⊢ dom {〈𝐴, 𝐵〉} ⊆ {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 {csn 4569 〈cop 4575 dom cdm 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-dm 5567 |
This theorem is referenced by: snopsuppss 7847 setsres 16527 setscom 16529 setsid 16540 strle1 16594 ex-res 28222 bj-fununsn1 34537 mapfzcons1 39321 |
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