Theorem s1dmALT 14574

Description: Alternate version of s1dm 14573, 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	⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = {0})

Proof of Theorem s1dmALT

Step	Hyp	Ref	Expression
1		s1val 14563	. . 3 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
2	1	dmeqd 5869	. 2 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = dom {⟨0, 𝐴⟩})
3		dmsnopg 6186	. 2 ⊢ (𝐴 ∈ 𝑆 → dom {⟨0, 𝐴⟩} = {0})
4	2, 3	eqtrd 2764	1 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = {0})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4589  ⟨cop 4595  dom cdm 5638  0cc0 11068  ⟨“cs1 14560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-s1 14561

This theorem is referenced by: (None)