Theorem enssdomOLD 9032

Description: Obsolete version of enssdom 9031 as of 10-Feb-2026. (Contributed by NM, 31-Mar-1998.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
enssdomOLD	⊢ ≈ ⊆ ≼

Proof of Theorem enssdomOLD

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relen 9006	. 2 ⊢ Rel ≈
2		f1of1 6845	. . . . 5 ⊢ (𝑓:𝑥⟶1-1onto𝑦 → 𝑓:𝑥⟶1-1𝑦)
3	2	eximi 1868	. . . 4 ⊢ (∃𝑓 𝑓:𝑥⟶1-1onto𝑦 → ∃𝑓 𝑓:𝑥⟶1-1𝑦)
4		opabidw 5502	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦} ↔ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦)
5		opabidw 5502	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1𝑦} ↔ ∃𝑓 𝑓:𝑥⟶1-1𝑦)
6	3, 4, 5	3imtr4i 295	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1𝑦})
7		df-en 9002	. . . 4 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦}
8	7	eleq2i 2856	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≈ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦})
9		df-dom 9003	. . . 4 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1𝑦}
10	9	eleq2i 2856	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≼ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1𝑦})
11	6, 8, 10	3imtr4i 295	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≈ → ⟨𝑥, 𝑦⟩ ∈ ≼ )
12	1, 11	relssi 5768	1 ⊢ ≈ ⊆ ≼

This proof depends on syntax axioms:  ∃wex 1812   ∈ wcel 2147   ⊆ wss 3900  ⟨cop 4838  {copab 5170  ⟶1-1wf1 6555  ⟶1-1ontowf1o 6557   ≈ cen 8998   ≼ cdom 8999

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661  df-rel 5662  df-f1o 6565  df-en 9002  df-dom 9003

This theorem is used by: (None)