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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.
StepHypRef Expression
1 relen 7945 . 2 Rel ≈
2 f1of1 6123 . . . . 5 (𝑓:𝑥1-1-onto𝑦𝑓:𝑥1-1𝑦)
32eximi 1760 . . . 4 (∃𝑓 𝑓:𝑥1-1-onto𝑦 → ∃𝑓 𝑓:𝑥1-1𝑦)
4 opabid 4972 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦} ↔ ∃𝑓 𝑓:𝑥1-1-onto𝑦)
5 opabid 4972 . . . 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 7941 . . . 4 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
87eleq2i 2691 . . 3 (⟨𝑥, 𝑦⟩ ∈ ≈ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦})
9 df-dom 7942 . . . 4 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
109eleq2i 2691 . . 3 (⟨𝑥, 𝑦⟩ ∈ ≼ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦})
116, 8, 103imtr4i 281 . 2 (⟨𝑥, 𝑦⟩ ∈ ≈ → ⟨𝑥, 𝑦⟩ ∈ ≼ )
121, 11relssi 5201 1 ≈ ⊆ ≼
Colors of variables: wff setvar class
Syntax hints:  wex 1702  wcel 1988  wss 3567  cop 4174  {copab 4703  1-1wf1 5873  1-1-ontowf1o 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
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