Theorem enssdom 8923

Description: Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) (Proof shortened by TM, 10-Feb-2026.)

Assertion

Ref	Expression
enssdom	⊢ ≈ ⊆ ≼

Proof of Theorem enssdom

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

Step	Hyp	Ref	Expression
1		f1of1 6779	. . . 4 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦)
2	1	eximi 1837	. . 3 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦)
3	2	ssopab2i 5505	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ⊆ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
4		df-en 8894	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
5		df-dom 8895	. 2 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
6	3, 4, 5	3sstr4i 3973	1 ⊢ ≈ ⊆ ≼

Syntax hints:  ∃wex 1781   ⊆ wss 3889  {copab 5147  –1-1→wf1 6495  –1-1-onto→wf1o 6497   ≈ cen 8890   ≼ cdom 8891

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-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-ss 3906  df-opab 5148  df-f1o 6505  df-en 8894  df-dom 8895

This theorem is referenced by:  dfdom2  8925  endom  8926