Theorem s1dmALT 14630

Description: Alternate version of s1dm 14629, 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 14619	. . 3 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
2	1	dmeqd 5898	. 2 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = dom {⟨0, 𝐴⟩})
3		dmsnopg 6215	. 2 ⊢ (𝐴 ∈ 𝑆 → dom {⟨0, 𝐴⟩} = {0})
4	2, 3	eqtrd 2769	1 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = {0})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  {csn 4608  ⟨cop 4614  dom cdm 5667  0cc0 11138  ⟨“cs1 14616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6495  df-fun 6544  df-fv 6550  df-s1 14617

This theorem is referenced by: (None)