Theorem ener 6952

Description: Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)

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 6912	. . . 4 ⊢ Rel ≈
2	1	a1i 9	. . 3 ⊢ (⊤ → Rel ≈ )
3		bren 6916	. . . . 5 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦)
4		f1ocnv 5596	. . . . . . 7 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → ◡𝑓:𝑦–1-1-onto→𝑥)
5		vex 2805	. . . . . . . 8 ⊢ 𝑦 ∈ V
6		vex 2805	. . . . . . . 8 ⊢ 𝑥 ∈ V
7		f1oen2g 6927	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ ◡𝑓:𝑦–1-1-onto→𝑥) → 𝑦 ≈ 𝑥)
8	5, 6, 7	mp3an12 1363	. . . . . . 7 ⊢ (◡𝑓:𝑦–1-1-onto→𝑥 → 𝑦 ≈ 𝑥)
9	4, 8	syl 14	. . . . . 6 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥)
10	9	exlimiv 1646	. . . . 5 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥)
11	3, 10	sylbi 121	. . . 4 ⊢ (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥)
12	11	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑥 ≈ 𝑦) → 𝑦 ≈ 𝑥)
13		bren 6916	. . . . 5 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1-onto→𝑦)
14		bren 6916	. . . . 5 ⊢ (𝑦 ≈ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧)
15		eeanv 1985	. . . . . 6 ⊢ (∃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))
16		f1oco 5606	. . . . . . . . 9 ⊢ ((𝑓:𝑦–1-1-onto→𝑧 ∧ 𝑔:𝑥–1-1-onto→𝑦) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧)
17	16	ancoms 268	. . . . . . . 8 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧)
18		vex 2805	. . . . . . . . 9 ⊢ 𝑧 ∈ V
19		f1oen2g 6927	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
20	6, 18, 19	mp3an12 1363	. . . . . . . 8 ⊢ ((𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧 → 𝑥 ≈ 𝑧)
21	17, 20	syl 14	. . . . . . 7 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
22	21	exlimivv 1945	. . . . . 6 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
23	15, 22	sylbir 135	. . . . 5 ⊢ ((∃𝑔 𝑔:𝑥–1-1-onto→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
24	13, 14, 23	syl2anb 291	. . . 4 ⊢ ((𝑥 ≈ 𝑦 ∧ 𝑦 ≈ 𝑧) → 𝑥 ≈ 𝑧)
25	24	adantl 277	. . 3 ⊢ ((⊤ ∧ (𝑥 ≈ 𝑦 ∧ 𝑦 ≈ 𝑧)) → 𝑥 ≈ 𝑧)
26	6	enref 6937	. . . . 5 ⊢ 𝑥 ≈ 𝑥
27	6, 26	2th 174	. . . 4 ⊢ (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥)
28	27	a1i 9	. . 3 ⊢ (⊤ → (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥))
29	2, 12, 25, 28	iserd 6727	. 2 ⊢ (⊤ → ≈ Er V)
30	29	mptru 1406	1 ⊢ ≈ Er V

Syntax hints:   ∧ wa 104   ↔ wb 105  ⊤wtru 1398  ∃wex 1540   ∈ wcel 2202  Vcvv 2802   class class class wbr 4088  ^◡ccnv 4724   ∘ ccom 4729  Rel wrel 4730  –1-1-onto→wf1o 5325   Er wer 6698   ≈ cen 6906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-er 6701  df-en 6909

This theorem is referenced by:  ensymb  6953  entr  6957