Theorem enssdom 8982

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 6816	. . . 4 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦)
2	1	eximi 1868	. . 3 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦)
3	2	ssopab2i 5529	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ⊆ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
4		df-en 8953	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
5		df-dom 8954	. 2 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
6	3, 4, 5	3sstr4i 3982	1 ⊢ ≈ ⊆ ≼

This proof depends on syntax axioms:  ∃wex 1812   ⊆ wss 3899  {copab 5167  –1-1→wf1 6530  –1-1-onto→wf1o 6532   ≈ cen 8949   ≼ cdom 8950

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 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-ss 3916  df-opab 5168  df-f1o 6540  df-en 8953  df-dom 8954

This theorem is used by:  dfdom2  8984  endom  8985