Theorem ener 8952

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 8902	. 2 ⊢ Rel ≈
2		bren 8907	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦)
3		vex 3446	. . . . 5 ⊢ 𝑦 ∈ V
4		vex 3446	. . . . 5 ⊢ 𝑥 ∈ V
5		f1ocnv 6796	. . . . 5 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → ◡𝑓:𝑦–1-1-onto→𝑥)
6		f1oen2g 8919	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ ◡𝑓:𝑦–1-1-onto→𝑥) → 𝑦 ≈ 𝑥)
7	3, 4, 5, 6	mp3an12i 1468	. . . 4 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥)
8	7	exlimiv 1932	. . 3 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥)
9	2, 8	sylbi 217	. 2 ⊢ (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥)
10		bren 8907	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1-onto→𝑦)
11		bren 8907	. . 3 ⊢ (𝑦 ≈ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧)
12		exdistrv 1957	. . . 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 3446	. . . . . 6 ⊢ 𝑧 ∈ V
14		f1oco 6807	. . . . . . 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 8919	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
17	4, 13, 15, 16	mp3an12i 1468	. . . . 5 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
18	17	exlimivv 1934	. . . 4 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
19	12, 18	sylbir 235	. . 3 ⊢ ((∃𝑔 𝑔:𝑥–1-1-onto→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
20	10, 11, 19	syl2anb 599	. 2 ⊢ ((𝑥 ≈ 𝑦 ∧ 𝑦 ≈ 𝑧) → 𝑥 ≈ 𝑧)
21	4	enref 8936	. . 3 ⊢ 𝑥 ≈ 𝑥
22	4, 21	2th 264	. 2 ⊢ (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥)
23	1, 9, 20, 22	iseri 8675	1 ⊢ ≈ Er V

Syntax hints:   ∧ wa 395  ∃wex 1781   ∈ wcel 2114  Vcvv 3442   class class class wbr 5100  ^◡ccnv 5633   ∘ ccom 5638  –1-1-onto→wf1o 6501   Er wer 8644   ≈ cen 8894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-er 8647  df-en 8898

This theorem is referenced by:  ensymb  8953  entr  8957