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Theorem enssdom 8528
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 8508 . 2 Rel ≈
2 f1of1 6609 . . . . 5 (𝑓:𝑥1-1-onto𝑦𝑓:𝑥1-1𝑦)
32eximi 1831 . . . 4 (∃𝑓 𝑓:𝑥1-1-onto𝑦 → ∃𝑓 𝑓:𝑥1-1𝑦)
4 opabidw 5405 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦} ↔ ∃𝑓 𝑓:𝑥1-1-onto𝑦)
5 opabidw 5405 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦} ↔ ∃𝑓 𝑓:𝑥1-1𝑦)
63, 4, 53imtr4i 294 . . 3 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦})
7 df-en 8504 . . . 4 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
87eleq2i 2904 . . 3 (⟨𝑥, 𝑦⟩ ∈ ≈ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦})
9 df-dom 8505 . . . 4 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
109eleq2i 2904 . . 3 (⟨𝑥, 𝑦⟩ ∈ ≼ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦})
116, 8, 103imtr4i 294 . 2 (⟨𝑥, 𝑦⟩ ∈ ≈ → ⟨𝑥, 𝑦⟩ ∈ ≼ )
121, 11relssi 5655 1 ≈ ⊆ ≼
Colors of variables: wff setvar class
Syntax hints:  wex 1776  wcel 2110  wss 3936  cop 4567  {copab 5121  1-1wf1 6347  1-1-ontowf1o 6349  cen 8500  cdom 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5122  df-xp 5556  df-rel 5557  df-f1o 6357  df-en 8504  df-dom 8505
This theorem is referenced by:  dfdom2  8529  endom  8530
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