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Mirrors > Home > MPE Home > Th. List > s1dmALT | Structured version Visualization version GIF version |
Description: Alternate version of s1dm 14596, having a shorter proof, but requiring that 𝐴 is a set. (Contributed by AV, 9-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
s1dmALT | ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = {0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1val 14586 | . . 3 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩}) | |
2 | 1 | dmeqd 5910 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = dom {⟨0, 𝐴⟩}) |
3 | dmsnopg 6220 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom {⟨0, 𝐴⟩} = {0}) | |
4 | 2, 3 | eqtrd 2767 | 1 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = {0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4630 ⟨cop 4636 dom cdm 5680 0cc0 11144 ⟨“cs1 14583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-iota 6503 df-fun 6553 df-fv 6559 df-s1 14584 |
This theorem is referenced by: (None) |
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