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Theorem enssdom 8765
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 8738 . 2 Rel ≈
2 f1of1 6715 . . . . 5 (𝑓:𝑥1-1-onto𝑦𝑓:𝑥1-1𝑦)
32eximi 1837 . . . 4 (∃𝑓 𝑓:𝑥1-1-onto𝑦 → ∃𝑓 𝑓:𝑥1-1𝑦)
4 opabidw 5437 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦} ↔ ∃𝑓 𝑓:𝑥1-1-onto𝑦)
5 opabidw 5437 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦} ↔ ∃𝑓 𝑓:𝑥1-1𝑦)
63, 4, 53imtr4i 292 . . 3 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦})
7 df-en 8734 . . . 4 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
87eleq2i 2830 . . 3 (⟨𝑥, 𝑦⟩ ∈ ≈ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦})
9 df-dom 8735 . . . 4 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
109eleq2i 2830 . . 3 (⟨𝑥, 𝑦⟩ ∈ ≼ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦})
116, 8, 103imtr4i 292 . 2 (⟨𝑥, 𝑦⟩ ∈ ≈ → ⟨𝑥, 𝑦⟩ ∈ ≼ )
121, 11relssi 5697 1 ≈ ⊆ ≼
Colors of variables: wff setvar class
Syntax hints:  wex 1782  wcel 2106  wss 3887  cop 4567  {copab 5136  1-1wf1 6430  1-1-ontowf1o 6432  cen 8730  cdom 8731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-xp 5595  df-rel 5596  df-f1o 6440  df-en 8734  df-dom 8735
This theorem is referenced by:  dfdom2  8766  endom  8767
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