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Mirrors > Home > ILE Home > Th. List > en2d | 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 |
---|---|
en2d.1 | ⊢ (𝜑 → 𝐴 ∈ V) |
en2d.2 | ⊢ (𝜑 → 𝐵 ∈ V) |
en2d.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ V)) |
en2d.4 | ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ V)) |
en2d.5 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) |
Ref | Expression |
---|---|
en2d | ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) | |
2 | en2d.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) | |
3 | eqid 2117 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
4 | en2d.3 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ V)) | |
5 | 4 | imp 123 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ V) |
6 | en2d.4 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ V)) | |
7 | 6 | imp 123 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V) |
8 | en2d.5 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) | |
9 | 3, 5, 7, 8 | f1od 5941 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) |
10 | f1oen2g 6617 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
11 | 1, 2, 9, 10 | syl3anc 1201 | 1 ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 Vcvv 2660 class class class wbr 3899 ↦ cmpt 3959 –1-1-onto→wf1o 5092 ≈ cen 6600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-en 6603 |
This theorem is referenced by: en2i 6632 map1 6674 |
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