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Theorem enssdom 6608
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 6590 . 2 Rel ≈
2 f1of1 5320 . . . . 5 (𝑓:𝑥1-1-onto𝑦𝑓:𝑥1-1𝑦)
32eximi 1560 . . . 4 (∃𝑓 𝑓:𝑥1-1-onto𝑦 → ∃𝑓 𝑓:𝑥1-1𝑦)
4 opabid 4137 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦} ↔ ∃𝑓 𝑓:𝑥1-1-onto𝑦)
5 opabid 4137 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦} ↔ ∃𝑓 𝑓:𝑥1-1𝑦)
63, 4, 53imtr4i 200 . . 3 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦})
7 df-en 6587 . . . 4 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
87eleq2i 2179 . . 3 (⟨𝑥, 𝑦⟩ ∈ ≈ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦})
9 df-dom 6588 . . . 4 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
109eleq2i 2179 . . 3 (⟨𝑥, 𝑦⟩ ∈ ≼ ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦})
116, 8, 103imtr4i 200 . 2 (⟨𝑥, 𝑦⟩ ∈ ≈ → ⟨𝑥, 𝑦⟩ ∈ ≼ )
121, 11relssi 4588 1 ≈ ⊆ ≼
Colors of variables: wff set class
Syntax hints:  wex 1449  wcel 1461  wss 3035  cop 3494  {copab 3946  1-1wf1 5076  1-1-ontowf1o 5078  cen 6584  cdom 6585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-opab 3948  df-xp 4503  df-rel 4504  df-f1o 5086  df-en 6587  df-dom 6588
This theorem is referenced by:  endom  6609
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