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Theorem breng 8993
Description: Equinumerosity relation. This variation of bren 8994 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 8994. (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 6838 . . 3 (𝑥 = 𝐴 → (𝑓:𝑥1-1-onto𝑦𝑓:𝐴1-1-onto𝑦))
21exbidv 1919 . 2 (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥1-1-onto𝑦 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝑦))
3 f1oeq3 6839 . . 3 (𝑦 = 𝐵 → (𝑓:𝐴1-1-onto𝑦𝑓:𝐴1-1-onto𝐵))
43exbidv 1919 . 2 (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴1-1-onto𝑦 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
5 df-en 8985 . 2 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
62, 4, 5brabg 5549 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106   class class class wbr 5148  1-1-ontowf1o 6562  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-en 8985
This theorem is referenced by:  bren  8994  f1oen4g  9004  en0  9057  en0r  9059  ensn1  9060  en1  9063  en2sn  9080  en2prd  9087  rexdif1en  9197  snnen2o  9271  1sdom2dom  9281  clnbgr3stgrgrlic  47915
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