Theorem enssdom 7965

Description: Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)

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		relen 7945	. 2 ⊢ Rel ≈
2		f1of1 6123	. . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦)
3	2	eximi 1760	. . . 4 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦)
4		opabid 4972	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦)
5		opabid 4972	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} ↔ ∃𝑓 𝑓:𝑥–1-1→𝑦)
6	3, 4, 5	3imtr4i 281	. . 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 7941	. . . 4 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
8	7	eleq2i 2691	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≈ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦})
9		df-dom 7942	. . . 4 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
10	9	eleq2i 2691	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≼ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦})
11	6, 8, 10	3imtr4i 281	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ≈ → ⟨𝑥, 𝑦⟩ ∈ ≼ )
12	1, 11	relssi 5201	1 ⊢ ≈ ⊆ ≼

Syntax hints:  ∃wex 1702   ∈ wcel 1988   ⊆ wss 3567  ⟨cop 4174  {copab 4703  –1-1→wf1 5873  –1-1-onto→wf1o 5875   ≈ cen 7937   ≼ cdom 7938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-xp 5110  df-rel 5111  df-f1o 5883  df-en 7941  df-dom 7942

This theorem is referenced by:  dfdom2  7966  endom  7967