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Mirrors > Home > MPE Home > Th. List > s1dmALT | Structured version Visualization version GIF version |
Description: Alternate version of s1dm 14503, 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 14493 | . . 3 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩}) | |
2 | 1 | dmeqd 5866 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = dom {⟨0, 𝐴⟩}) |
3 | dmsnopg 6170 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom {⟨0, 𝐴⟩} = {0}) | |
4 | 2, 3 | eqtrd 2777 | 1 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = {0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {csn 4591 ⟨cop 4597 dom cdm 5638 0cc0 11058 ⟨“cs1 14490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-s1 14491 |
This theorem is referenced by: (None) |
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