![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > breng | Structured version Visualization version GIF version |
Description: Equinumerosity relation. This variation of bren 8890 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 8890. (Revised by BTernaryTau, 23-Sep-2024.) |
Ref | Expression |
---|---|
breng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
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 1924 | . 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 1924 | . 2 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1-onto→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
5 | df-en 8881 | . 2 ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | |
6 | 2, 4, 5 | brabg 5495 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 class class class wbr 5104 –1-1-onto→wf1o 6493 ≈ cen 8877 |
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 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
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 2714 df-cleq 2728 df-clel 2814 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-en 8881 |
This theorem is referenced by: bren 8890 f1oen4g 8901 en0 8954 en0r 8957 ensn1 8958 en1 8962 en2sn 8982 en2prd 8989 rexdif1en 9099 snnen2o 9178 1sdom2dom 9188 |
Copyright terms: Public domain | W3C validator |