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Theorem enssdom 8022
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.
StepHypRef Expression
1 relen 8002 . 2 Rel ≈
2 f1of1 6174 . . . . 5 (𝑓:𝑥1-1-onto𝑦𝑓:𝑥1-1𝑦)
32eximi 1802 . . . 4 (∃𝑓 𝑓:𝑥1-1-onto𝑦 → ∃𝑓 𝑓:𝑥1-1𝑦)
4 opabid 5011 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦} ↔ ∃𝑓 𝑓:𝑥1-1-onto𝑦)
5 opabid 5011 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦} ↔ ∃𝑓 𝑓:𝑥1-1𝑦)
63, 4, 53imtr4i 281 . . 3 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦})
7 df-en 7998 . . . 4 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
87eleq2i 2722 . . 3 (⟨𝑥, 𝑦⟩ ∈ ≈ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦})
9 df-dom 7999 . . . 4 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
109eleq2i 2722 . . 3 (⟨𝑥, 𝑦⟩ ∈ ≼ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦})
116, 8, 103imtr4i 281 . 2 (⟨𝑥, 𝑦⟩ ∈ ≈ → ⟨𝑥, 𝑦⟩ ∈ ≼ )
121, 11relssi 5245 1 ≈ ⊆ ≼
Colors of variables: wff setvar class
Syntax hints:  wex 1744  wcel 2030  wss 3607  cop 4216  {copab 4745  1-1wf1 5923  1-1-ontowf1o 5925  cen 7994  cdom 7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150  df-f1o 5933  df-en 7998  df-dom 7999
This theorem is referenced by:  dfdom2  8023  endom  8024
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