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Theorem s1dmALT 14563
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.)
Assertion
Ref Expression
s1dmALT (𝐴𝑆 → dom ⟨“𝐴”⟩ = {0})

Proof of Theorem s1dmALT
StepHypRef Expression
1 s1val 14552 . . 3 (𝐴𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
21dmeqd 5898 . 2 (𝐴𝑆 → dom ⟨“𝐴”⟩ = dom {⟨0, 𝐴⟩})
3 dmsnopg 6205 . 2 (𝐴𝑆 → dom {⟨0, 𝐴⟩} = {0})
42, 3eqtrd 2766 1 (𝐴𝑆 → 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)
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