Theorem breng 9007

Description: Equinumerosity relation. This variation of bren 9008 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 9008. (Revised by BTernaryTau, 23-Sep-2024.)

Assertion

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

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 6832	. . 3 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥⟶1-1onto𝑦 ↔ 𝑓:𝐴⟶1-1onto𝑦))
2	1	exbidv 1956	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥⟶1-1onto𝑦 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝑦))
3		f1oeq3 6833	. . 3 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴⟶1-1onto𝑦 ↔ 𝑓:𝐴⟶1-1onto𝐵))
4	3	exbidv 1956	. 2 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴⟶1-1onto𝑦 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝐵))
5		df-en 8999	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦}
6	2, 4, 5	brabg 5518	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   class class class wbr 5106  ⟶1-1ontowf1o 6554   ≈ cen 8995

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-en 8999

This theorem is used by:  bren  9008  f1oen4g  9016  en0  9070  en0r  9072  ensn1  9073  en1  9076  en2sn  9094  en2prd  9100  rexdif1en  9201  snnen2o  9261  1sdom2dom  9270  clnbgr3stgrgrlic  49321