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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 8941 | . 2 ⊢ Rel ≈ | |
2 | bren 8946 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦) | |
3 | vex 3470 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3470 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | f1ocnv 6836 | . . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → ◡𝑓:𝑦–1-1-onto→𝑥) | |
6 | f1oen2g 8961 | . . . . 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 1925 | . . 3 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥) |
9 | 2, 8 | sylbi 216 | . 2 ⊢ (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥) |
10 | bren 8946 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1-onto→𝑦) | |
11 | bren 8946 | . . 3 ⊢ (𝑦 ≈ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧) | |
12 | exdistrv 1951 | . . . 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 3470 | . . . . . 6 ⊢ 𝑧 ∈ V | |
14 | f1oco 6847 | . . . . . . 7 ⊢ ((𝑓:𝑦–1-1-onto→𝑧 ∧ 𝑔:𝑥–1-1-onto→𝑦) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) | |
15 | 14 | ancoms 458 | . . . . . 6 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) |
16 | f1oen2g 8961 | . . . . . 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 1927 | . . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) |
19 | 12, 18 | sylbir 234 | . . 3 ⊢ ((∃𝑔 𝑔:𝑥–1-1-onto→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) |
20 | 10, 11, 19 | syl2anb 597 | . 2 ⊢ ((𝑥 ≈ 𝑦 ∧ 𝑦 ≈ 𝑧) → 𝑥 ≈ 𝑧) |
21 | 4 | enref 8978 | . . 3 ⊢ 𝑥 ≈ 𝑥 |
22 | 4, 21 | 2th 264 | . 2 ⊢ (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥) |
23 | 1, 9, 20, 22 | iseri 8727 | 1 ⊢ ≈ Er V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∃wex 1773 ∈ wcel 2098 Vcvv 3466 class class class wbr 5139 ◡ccnv 5666 ∘ ccom 5671 –1-1-onto→wf1o 6533 Er wer 8697 ≈ cen 8933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-er 8700 df-en 8937 |
This theorem is referenced by: ensymb 8995 entr 8999 |
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