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Theorem s1dmALT 14597
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.)
Assertion
Ref Expression
s1dmALT (𝐴𝑆 → dom ⟨“𝐴”⟩ = {0})

Proof of Theorem s1dmALT
StepHypRef Expression
1 s1val 14586 . . 3 (𝐴𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
21dmeqd 5910 . 2 (𝐴𝑆 → dom ⟨“𝐴”⟩ = dom {⟨0, 𝐴⟩})
3 dmsnopg 6220 . 2 (𝐴𝑆 → dom {⟨0, 𝐴⟩} = {0})
42, 3eqtrd 2767 1 (𝐴𝑆 → 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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