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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 8227 | . 2 ⊢ Rel ≈ | |
2 | bren 8231 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦) | |
3 | vex 3417 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3417 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | f1ocnv 6390 | . . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → ◡𝑓:𝑦–1-1-onto→𝑥) | |
6 | f1oen2g 8239 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ ◡𝑓:𝑦–1-1-onto→𝑥) → 𝑦 ≈ 𝑥) | |
7 | 3, 4, 5, 6 | mp3an12i 1593 | . . . 4 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥) |
8 | 7 | exlimiv 2029 | . . 3 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥) |
9 | 2, 8 | sylbi 209 | . 2 ⊢ (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥) |
10 | bren 8231 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1-onto→𝑦) | |
11 | bren 8231 | . . 3 ⊢ (𝑦 ≈ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧) | |
12 | exdistrv 2054 | . . . 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 3417 | . . . . . 6 ⊢ 𝑧 ∈ V | |
14 | f1oco 6400 | . . . . . . 7 ⊢ ((𝑓:𝑦–1-1-onto→𝑧 ∧ 𝑔:𝑥–1-1-onto→𝑦) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) | |
15 | 14 | ancoms 452 | . . . . . 6 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) |
16 | f1oen2g 8239 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) | |
17 | 4, 13, 15, 16 | mp3an12i 1593 | . . . . 5 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) |
18 | 17 | exlimivv 2031 | . . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) |
19 | 12, 18 | sylbir 227 | . . 3 ⊢ ((∃𝑔 𝑔:𝑥–1-1-onto→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) |
20 | 10, 11, 19 | syl2anb 591 | . 2 ⊢ ((𝑥 ≈ 𝑦 ∧ 𝑦 ≈ 𝑧) → 𝑥 ≈ 𝑧) |
21 | 4 | enref 8255 | . . 3 ⊢ 𝑥 ≈ 𝑥 |
22 | 4, 21 | 2th 256 | . 2 ⊢ (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥) |
23 | 1, 9, 20, 22 | iseri 8036 | 1 ⊢ ≈ Er V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∃wex 1878 ∈ wcel 2164 Vcvv 3414 class class class wbr 4873 ◡ccnv 5341 ∘ ccom 5346 –1-1-onto→wf1o 6122 Er wer 8006 ≈ cen 8219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-er 8009 df-en 8223 |
This theorem is referenced by: ensymb 8270 entr 8274 |
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