Theorem breng 8904

Description: Equinumerosity relation. This variation of bren 8905 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 8905. (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 6771	. . 3 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝑦))
2	1	exbidv 1921	. 2 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑦))
3		f1oeq3 6772	. . 3 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1-onto→𝑦 ↔ 𝑓:𝐴–1-1-onto→𝐵))
4	3	exbidv 1921	. 2 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
5		df-en 8896	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
6	2, 4, 5	brabg 5494	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   class class class wbr 5102  –1-1-onto→wf1o 6498   ≈ cen 8892

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-en 8896

This theorem is referenced by:  bren  8905  f1oen4g  8913  en0  8966  en0r  8968  ensn1  8969  en1  8972  en2sn  8989  en2prd  8996  rexdif1en  9099  snnen2o  9161  1sdom2dom  9170  clnbgr3stgrgrlic  48004