Theorem s1dmALT 14584

Description: Alternate version of s1dm 14583, 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 14573	. . 3 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
2	1	dmeqd 5877	. 2 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = dom {⟨0, 𝐴⟩})
3		dmsnopg 6194	. 2 ⊢ (𝐴 ∈ 𝑆 → dom {⟨0, 𝐴⟩} = {0})
4	2, 3	eqtrd 2765	1 ⊢ (𝐴 ∈ 𝑆 → dom ⟨“𝐴”⟩ = {0})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4597  ⟨cop 4603  dom cdm 5646  0cc0 11086  ⟨“cs1 14570

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 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-iota 6472  df-fun 6521  df-fv 6527  df-s1 14571

This theorem is referenced by: (None)