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Theorem breng 8994
Description: Equinumerosity relation. This variation of bren 8995 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 8995. (Revised by BTernaryTau, 23-Sep-2024.)
Assertion
Ref Expression
breng ((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hints:   𝑉(𝑓)   𝑊(𝑓)

Proof of Theorem breng
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oeq2 6837 . . 3 (𝑥 = 𝐴 → (𝑓:𝑥1-1-onto𝑦𝑓:𝐴1-1-onto𝑦))
21exbidv 1921 . 2 (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥1-1-onto𝑦 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝑦))
3 f1oeq3 6838 . . 3 (𝑦 = 𝐵 → (𝑓:𝐴1-1-onto𝑦𝑓:𝐴1-1-onto𝐵))
43exbidv 1921 . 2 (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴1-1-onto𝑦 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
5 df-en 8986 . 2 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
62, 4, 5brabg 5544 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108   class class class wbr 5143  1-1-ontowf1o 6560  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986
This theorem is referenced by:  bren  8995  f1oen4g  9005  en0  9058  en0r  9060  ensn1  9061  en1  9064  en2sn  9081  en2prd  9088  rexdif1en  9198  snnen2o  9273  1sdom2dom  9283  clnbgr3stgrgrlic  47979
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