Theorem enssdom 9041

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 6853	. . . 4 ⊢ (𝑓:𝑥⟶1-1onto𝑦 → 𝑓:𝑥⟶1-1𝑦)
2	1	eximi 1868	. . 3 ⊢ (∃𝑓 𝑓:𝑥⟶1-1onto𝑦 → ∃𝑓 𝑓:𝑥⟶1-1𝑦)
3	2	ssopab2i 5532	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦} ⊆ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1𝑦}
4		df-en 9012	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦}
5		df-dom 9013	. 2 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1𝑦}
6	3, 4, 5	3sstr4i 3964	1 ⊢ ≈ ⊆ ≼

This proof depends on syntax axioms:  ∃wex 1812   ⊆ wss 3900  {copab 5173  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565   ≈ cen 9008   ≼ cdom 9009

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-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-ss 3901  df-opab 5174  df-f1o 6573  df-en 9012  df-dom 9013

This theorem is used by:  dfdom2  9043  endom  9044