Theorem ener 6833

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 6798	. . . 4 ⊢ Rel ≈
2	1	a1i 9	. . 3 ⊢ (⊤ → Rel ≈ )
3		bren 6801	. . . . 5 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦)
4		f1ocnv 5513	. . . . . . 7 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → ◡𝑓:𝑦–1-1-onto→𝑥)
5		vex 2763	. . . . . . . 8 ⊢ 𝑦 ∈ V
6		vex 2763	. . . . . . . 8 ⊢ 𝑥 ∈ V
7		f1oen2g 6809	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ ◡𝑓:𝑦–1-1-onto→𝑥) → 𝑦 ≈ 𝑥)
8	5, 6, 7	mp3an12 1338	. . . . . . 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 1609	. . . . 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 6801	. . . . 5 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1-onto→𝑦)
14		bren 6801	. . . . 5 ⊢ (𝑦 ≈ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧)
15		eeanv 1948	. . . . . 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 5523	. . . . . . . . 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 2763	. . . . . . . . 9 ⊢ 𝑧 ∈ V
19		f1oen2g 6809	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
20	6, 18, 19	mp3an12 1338	. . . . . . . 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 1908	. . . . . 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 6819	. . . . 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 6613	. 2 ⊢ (⊤ → ≈ Er V)
30	29	mptru 1373	1 ⊢ ≈ Er V

Syntax hints:   ∧ wa 104   ↔ wb 105  ⊤wtru 1365  ∃wex 1503   ∈ wcel 2164  Vcvv 2760   class class class wbr 4029  ^◡ccnv 4658   ∘ ccom 4663  Rel wrel 4664  –1-1-onto→wf1o 5253   Er wer 6584   ≈ cen 6792

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-er 6587  df-en 6795

This theorem is referenced by:  ensymb  6834  entr  6838