Theorem breng 8964

Description: Equinumerosity relation. This variation of bren 8965 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 8965. (Revised by BTernaryTau, 23-Sep-2024.)

Assertion

Ref	Expression
breng	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))

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.

Step	Hyp	Ref	Expression
1		f1oeq2 6810	. . 3 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝑦))
2	1	exbidv 1954	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑦))
3		f1oeq3 6811	. . 3 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝐵))
4	3	exbidv 1954	. 2 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
5		df-en 8956	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
6	2, 4, 5	brabg 5522	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   class class class wbr 5107  –1-1-onto→wf1o 6536   ≈ cen 8952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-en 8956

This theorem is used by:  bren  8965  f1oen4g  8973  en0  9027  en0r  9029  ensn1  9030  en1  9033  en2sn  9051  en2prd  9057  rexdif1en  9158  snnen2o  9218  1sdom2dom  9227  clnbgr3stgrgrlic  48921