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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 8960 | . 2 ⊢ Rel ≈ | |
2 | bren 8965 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦) | |
3 | vex 3473 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | vex 3473 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | f1ocnv 6845 | . . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → ◡𝑓:𝑦–1-1-onto→𝑥) | |
6 | f1oen2g 8980 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ ◡𝑓:𝑦–1-1-onto→𝑥) → 𝑦 ≈ 𝑥) | |
7 | 3, 4, 5, 6 | mp3an12i 1462 | . . . 4 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥) |
8 | 7 | exlimiv 1926 | . . 3 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥) |
9 | 2, 8 | sylbi 216 | . 2 ⊢ (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥) |
10 | bren 8965 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1-onto→𝑦) | |
11 | bren 8965 | . . 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 3473 | . . . . . 6 ⊢ 𝑧 ∈ V | |
14 | f1oco 6856 | . . . . . . 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 8980 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) | |
17 | 4, 13, 15, 16 | mp3an12i 1462 | . . . . 5 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧) |
18 | 17 | exlimivv 1928 | . . . 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 8997 | . . 3 ⊢ 𝑥 ≈ 𝑥 |
22 | 4, 21 | 2th 264 | . 2 ⊢ (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥) |
23 | 1, 9, 20, 22 | iseri 8745 | 1 ⊢ ≈ Er V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∃wex 1774 ∈ wcel 2099 Vcvv 3469 class class class wbr 5142 ◡ccnv 5671 ∘ ccom 5676 –1-1-onto→wf1o 6541 Er wer 8715 ≈ cen 8952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-er 8718 df-en 8956 |
This theorem is referenced by: ensymb 9014 entr 9018 |
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