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Mirrors > Home > MPE Home > Th. List > en3d | Structured version Visualization version 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.) (Revised by AV, 4-Aug-2024.) |
Ref | Expression |
---|---|
en3d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
en3d.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
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 ∈ 𝑊) | |
3 | eqid 2738 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
4 | en3d.3 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) | |
5 | 4 | imp 410 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
6 | en3d.4 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴)) | |
7 | 6 | imp 410 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴) |
8 | en3d.5 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))) | |
9 | 8 | imp 410 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) |
10 | 3, 5, 7, 9 | f1o2d 7409 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) |
11 | f1oen2g 8565 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
12 | 1, 2, 10, 11 | syl3anc 1372 | 1 ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 class class class wbr 5027 ↦ cmpt 5107 –1-1-onto→wf1o 6332 ≈ cen 8545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-en 8549 |
This theorem is referenced by: en3i 8587 fundmen 8623 mapen 8724 mapxpen 8726 mapunen 8729 ssenen 8734 fzen 13008 hashbclem 13895 hashfacen 13897 hashfacenOLD 13898 hashf1lem1 13899 hashf1lem1OLD 13900 hashdvds 16205 |
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