Theorem ener 9056

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 9006	. 2 ⊢ Rel ≈
2		bren 9011	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦)
3		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
4		vex 3458	. . . . 5 ⊢ 𝑥 ∈ V
5		f1ocnv 6859	. . . . 5 ⊢ (𝑓:𝑥⟶1-1onto𝑦 → ◡𝑓:𝑦⟶1-1onto𝑥)
6		f1oen2g 9023	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ ◡𝑓:𝑦⟶1-1onto𝑥) → 𝑦 ≈ 𝑥)
7	3, 4, 5, 6	mp3an12i 1495	. . . 4 ⊢ (𝑓:𝑥⟶1-1onto𝑦 → 𝑦 ≈ 𝑥)
8	7	exlimiv 1965	. . 3 ⊢ (∃𝑓 𝑓:𝑥⟶1-1onto𝑦 → 𝑦 ≈ 𝑥)
9	2, 8	sylbi 220	. 2 ⊢ (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥)
10		bren 9011	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥⟶1-1onto𝑦)
11		bren 9011	. . 3 ⊢ (𝑦 ≈ 𝑧 ↔ ∃𝑓 𝑓:𝑦⟶1-1onto𝑧)
12		exdistrv 1990	. . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥⟶1-1onto𝑦 ∧ 𝑓:𝑦⟶1-1onto𝑧) ↔ (∃𝑔 𝑔:𝑥⟶1-1onto𝑦 ∧ ∃𝑓 𝑓:𝑦⟶1-1onto𝑧))
13		vex 3458	. . . . . 6 ⊢ 𝑧 ∈ V
14		f1oco 6870	. . . . . . 7 ⊢ ((𝑓:𝑦⟶1-1onto𝑧 ∧ 𝑔:𝑥⟶1-1onto𝑦) → (𝑓 ∘ 𝑔):𝑥⟶1-1onto𝑧)
15	14	ancoms 464	. . . . . 6 ⊢ ((𝑔:𝑥⟶1-1onto𝑦 ∧ 𝑓:𝑦⟶1-1onto𝑧) → (𝑓 ∘ 𝑔):𝑥⟶1-1onto𝑧)
16		f1oen2g 9023	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓 ∘ 𝑔):𝑥⟶1-1onto𝑧) → 𝑥 ≈ 𝑧)
17	4, 13, 15, 16	mp3an12i 1495	. . . . 5 ⊢ ((𝑔:𝑥⟶1-1onto𝑦 ∧ 𝑓:𝑦⟶1-1onto𝑧) → 𝑥 ≈ 𝑧)
18	17	exlimivv 1967	. . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥⟶1-1onto𝑦 ∧ 𝑓:𝑦⟶1-1onto𝑧) → 𝑥 ≈ 𝑧)
19	12, 18	sylbir 238	. . 3 ⊢ ((∃𝑔 𝑔:𝑥⟶1-1onto𝑦 ∧ ∃𝑓 𝑓:𝑦⟶1-1onto𝑧) → 𝑥 ≈ 𝑧)
20	10, 11, 19	syl2anb 610	. 2 ⊢ ((𝑥 ≈ 𝑦 ∧ 𝑦 ≈ 𝑧) → 𝑥 ≈ 𝑧)
21	4	enref 9040	. . 3 ⊢ 𝑥 ≈ 𝑥
22	4, 21	2th 267	. 2 ⊢ (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥)
23	1, 9, 20, 22	iseri 8773	1 ⊢ ≈ Er V

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106  ^◡ccnv 5654   ∘ ccom 5659  ⟶1-1ontowf1o 6557   Er wer 8742   ≈ cen 8998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-er 8745  df-en 9002

This theorem is used by:  ensymb  9057  entr  9061