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Mirrors > Home > MPE Home > Th. List > ener | Structured version Visualization version GIF version |
Description: Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof shortened by AV, 1-May-2021.) |
Ref | Expression |
---|---|
ener | ⊢ ≈ Er V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 8508 | . 2 ⊢ Rel ≈ | |
2 | bren 8512 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦) | |
3 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | f1ocnv 6621 | . . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → ◡𝑓:𝑦–1-1-onto→𝑥) | |
6 | f1oen2g 8520 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ ◡𝑓:𝑦–1-1-onto→𝑥) → 𝑦 ≈ 𝑥) | |
7 | 3, 4, 5, 6 | mp3an12i 1461 | . . . 4 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥) |
8 | 7 | exlimiv 1927 | . . 3 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥) |
9 | 2, 8 | sylbi 219 | . 2 ⊢ (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥) |
10 | bren 8512 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1-onto→𝑦) | |
11 | bren 8512 | . . 3 ⊢ (𝑦 ≈ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧) | |
12 | exdistrv 1952 | . . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) ↔ (∃𝑔 𝑔:𝑥–1-1-onto→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧)) | |
13 | vex 3497 | . . . . . 6 ⊢ 𝑧 ∈ V | |
14 | f1oco 6631 | . . . . . . 7 ⊢ ((𝑓:𝑦–1-1-onto→𝑧 ∧ 𝑔:𝑥–1-1-onto→𝑦) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) | |
15 | 14 | ancoms 461 | . . . . . 6 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) |
16 | f1oen2g 8520 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) | |
17 | 4, 13, 15, 16 | mp3an12i 1461 | . . . . 5 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) |
18 | 17 | exlimivv 1929 | . . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) |
19 | 12, 18 | sylbir 237 | . . 3 ⊢ ((∃𝑔 𝑔:𝑥–1-1-onto→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) |
20 | 10, 11, 19 | syl2anb 599 | . 2 ⊢ ((𝑥 ≈ 𝑦 ∧ 𝑦 ≈ 𝑧) → 𝑥 ≈ 𝑧) |
21 | 4 | enref 8536 | . . 3 ⊢ 𝑥 ≈ 𝑥 |
22 | 4, 21 | 2th 266 | . 2 ⊢ (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥) |
23 | 1, 9, 20, 22 | iseri 8310 | 1 ⊢ ≈ Er V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 ◡ccnv 5548 ∘ ccom 5553 –1-1-onto→wf1o 6348 Er wer 8280 ≈ cen 8500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-er 8283 df-en 8504 |
This theorem is referenced by: ensymb 8551 entr 8555 |
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