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Theorem breng 8931
Description: Equinumerosity relation. This variation of bren 8932 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 8932. (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 6809 . . 3 (𝑥 = 𝐴 → (𝑓:𝑥1-1-onto𝑦𝑓:𝐴1-1-onto𝑦))
21exbidv 1924 . 2 (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥1-1-onto𝑦 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝑦))
3 f1oeq3 6810 . . 3 (𝑦 = 𝐵 → (𝑓:𝐴1-1-onto𝑦𝑓:𝐴1-1-onto𝐵))
43exbidv 1924 . 2 (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴1-1-onto𝑦 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
5 df-en 8923 . 2 ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
62, 4, 5brabg 5532 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106   class class class wbr 5141  1-1-ontowf1o 6531  cen 8919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-en 8923
This theorem is referenced by:  bren  8932  f1oen4g  8943  en0  8996  en0r  8999  ensn1  9000  en1  9004  en2sn  9024  en2prd  9031  rexdif1en  9141  snnen2o  9220  1sdom2dom  9230
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