Theorem enssdomOLD 8926

Description: Obsolete version of enssdom 8925 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 8900	. 2 ⊢ Rel ≈
2		f1of1 6781	. . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦)
3	2	eximi 1837	. . . 4 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦)
4		opabidw 5480	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦)
5		opabidw 5480	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1→𝑦)
6	3, 4, 5	3imtr4i 292	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦})
7		df-en 8896	. . . 4 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
8	7	eleq2i 2829	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≈ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦})
9		df-dom 8897	. . . 4 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
10	9	eleq2i 2829	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≼ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦})
11	6, 8, 10	3imtr4i 292	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≈ → ⟨𝑥, 𝑦⟩ ∈ ≼ )
12	1, 11	relssi 5744	1 ⊢ ≈ ⊆ ≼

Syntax hints:  ∃wex 1781   ∈ wcel 2114   ⊆ wss 3903  ⟨cop 4588  {copab 5162  –1-1→wf1 6497  –1-1-onto→wf1o 6499   ≈ cen 8892   ≼ cdom 8893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5638  df-rel 5639  df-f1o 6507  df-en 8896  df-dom 8897

This theorem is referenced by: (None)