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Theorem s1dmALT 13699
Description: Alternate version of s1dm 13698, 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.)
Assertion
Ref Expression
s1dmALT (𝐴𝑆 → dom ⟨“𝐴”⟩ = {0})

Proof of Theorem s1dmALT
StepHypRef Expression
1 s1val 13688 . . 3 (𝐴𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
21dmeqd 5571 . 2 (𝐴𝑆 → dom ⟨“𝐴”⟩ = dom {⟨0, 𝐴⟩})
3 dmsnopg 5860 . 2 (𝐴𝑆 → dom {⟨0, 𝐴⟩} = {0})
42, 3eqtrd 2813 1 (𝐴𝑆 → dom ⟨“𝐴”⟩ = {0})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2106  {csn 4397  cop 4403  dom cdm 5355  0cc0 10272  ⟨“cs1 13685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-s1 13686
This theorem is referenced by: (None)
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