Theorem ener 8119

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.)

Assertion

Ref	Expression
ener	⊢ ≈ Er V

Proof of Theorem ener

Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relen 8077	. 2 ⊢ Rel ≈
2		bren 8081	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦)
3		vex 3307	. . . . 5 ⊢ 𝑦 ∈ V
4		vex 3307	. . . . 5 ⊢ 𝑥 ∈ V
5		f1ocnv 6262	. . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → ◡𝑓:𝑦–1-1-onto→𝑥)
6		f1oen2g 8089	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ ◡𝑓:𝑦–1-1-onto→𝑥) → 𝑦 ≈ 𝑥)
7	3, 4, 5, 6	mp3an12i 1541	. . . 4 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥)
8	7	exlimiv 1971	. . 3 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥)
9	2, 8	sylbi 207	. 2 ⊢ (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥)
10		bren 8081	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1-onto→𝑦)
11		bren 8081	. . 3 ⊢ (𝑦 ≈ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧)
12		eeanv 2291	. . . 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 3307	. . . . . 6 ⊢ 𝑧 ∈ V
14		f1oco 6272	. . . . . . 7 ⊢ ((𝑓:𝑦–1-1-onto→𝑧 ∧ 𝑔:𝑥–1-1-onto→𝑦) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧)
15	14	ancoms 468	. . . . . 6 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧)
16		f1oen2g 8089	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
17	4, 13, 15, 16	mp3an12i 1541	. . . . 5 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
18	17	exlimivv 1973	. . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
19	12, 18	sylbir 225	. . 3 ⊢ ((∃𝑔 𝑔:𝑥–1-1-onto→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
20	10, 11, 19	syl2anb 497	. 2 ⊢ ((𝑥 ≈ 𝑦 ∧ 𝑦 ≈ 𝑧) → 𝑥 ≈ 𝑧)
21	4	enref 8105	. . 3 ⊢ 𝑥 ≈ 𝑥
22	4, 21	2th 254	. 2 ⊢ (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥)
23	1, 9, 20, 22	iseri 7889	1 ⊢ ≈ Er V

Syntax hints:   ∧ wa 383  ∃wex 1817   ∈ wcel 2103  Vcvv 3304   class class class wbr 4760  ^◡ccnv 5217   ∘ ccom 5222  –1-1-onto→wf1o 6000   Er wer 7859   ≈ cen 8069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-er 7862  df-en 8073

This theorem is referenced by:  ensymb  8120  entr  8124