![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > s1dmALT | Structured version Visualization version GIF version |
Description: Alternate version of s1dm 14562, 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 14552 | . . 3 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩}) | |
2 | 1 | dmeqd 5898 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = dom {⟨0, 𝐴⟩}) |
3 | dmsnopg 6205 | . 2 ⊢ (𝐴 ∈ 𝑆 → dom {⟨0, 𝐴⟩} = {0}) | |
4 | 2, 3 | eqtrd 2766 | 1 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = {0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4623 ⟨cop 4629 dom cdm 5669 0cc0 11109 ⟨“cs1 14549 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-s1 14550 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |