![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > en3d | GIF version |
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) |
Ref | Expression |
---|---|
en3d.1 | ⊢ (𝜑 → 𝐴 ∈ V) |
en3d.2 | ⊢ (𝜑 → 𝐵 ∈ V) |
en3d.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
en3d.4 | ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴)) |
en3d.5 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))) |
Ref | Expression |
---|---|
en3d | ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en3d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) | |
2 | en3d.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) | |
3 | eqid 2088 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
4 | en3d.3 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) | |
5 | 4 | imp 122 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
6 | en3d.4 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴)) | |
7 | 6 | imp 122 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴) |
8 | en3d.5 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))) | |
9 | 8 | imp 122 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) |
10 | 3, 5, 7, 9 | f1o2d 5849 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) |
11 | f1oen2g 6472 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
12 | 1, 2, 10, 11 | syl3anc 1174 | 1 ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1289 ∈ wcel 1438 Vcvv 2619 class class class wbr 3845 ↦ cmpt 3899 –1-1-onto→wf1o 5014 ≈ cen 6455 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-en 6458 |
This theorem is referenced by: en3i 6488 fundmen 6523 mapen 6562 mapxpen 6564 ssenen 6567 fzen 9457 hashfacen 10241 hashdvds 11475 |
Copyright terms: Public domain | W3C validator |