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Theorem brdom2g 8946
Description: Dominance relation. This variation of brdomg 8947 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 8947. (Revised by BTernaryTau, 29-Nov-2024.)
Assertion
Ref Expression
brdom2g ((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hints:   𝑉(𝑓)   𝑊(𝑓)

Proof of Theorem brdom2g
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1eq2 6773 . . 3 (𝑥 = 𝐴 → (𝑓:𝑥1-1𝑦𝑓:𝐴1-1𝑦))
21exbidv 1916 . 2 (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥1-1𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝑦))
3 f1eq3 6774 . . 3 (𝑦 = 𝐵 → (𝑓:𝐴1-1𝑦𝑓:𝐴1-1𝐵))
43exbidv 1916 . 2 (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴1-1𝑦 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
5 df-dom 8936 . 2 ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
62, 4, 5brabg 5529 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098   class class class wbr 5138  1-1wf1 6530  cdom 8932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-fn 6536  df-f 6537  df-f1 6538  df-dom 8936
This theorem is referenced by:  brdomg  8947  brdomi  8949  f1dom4g  8956  0domg  9095
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