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Theorem s1dmALT 14312
Description: Alternate version of s1dm 14311, 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 14301 . . 3 (𝐴𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
21dmeqd 5813 . 2 (𝐴𝑆 → dom ⟨“𝐴”⟩ = dom {⟨0, 𝐴⟩})
3 dmsnopg 6115 . 2 (𝐴𝑆 → dom {⟨0, 𝐴⟩} = {0})
42, 3eqtrd 2780 1 (𝐴𝑆 → dom ⟨“𝐴”⟩ = {0})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  {csn 4567  cop 4573  dom cdm 5590  0cc0 10872  ⟨“cs1 14298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-iota 6390  df-fun 6434  df-fv 6440  df-s1 14299
This theorem is referenced by: (None)
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