Theorem ener 7056

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 7016	. . . 4 ⊢ Rel ≈
2	1	a1i 9	. . 3 ⊢ (⊤ → Rel ≈ )
3		bren 7020	. . . . 5 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦)
4		f1ocnv 5647	. . . . . . 7 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → ◡𝑓:𝑦–1-1-onto→𝑥)
5		vex 2824	. . . . . . . 8 ⊢ 𝑦 ∈ V
6		vex 2824	. . . . . . . 8 ⊢ 𝑥 ∈ V
7		f1oen2g 7031	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ ◡𝑓:𝑦–1-1-onto→𝑥) → 𝑦 ≈ 𝑥)
8	5, 6, 7	mp3an12 1368	. . . . . . 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 1651	. . . . 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 7020	. . . . 5 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1-onto→𝑦)
14		bren 7020	. . . . 5 ⊢ (𝑦 ≈ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧)
15		eeanv 1992	. . . . . 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 5657	. . . . . . . . 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 2824	. . . . . . . . 9 ⊢ 𝑧 ∈ V
19		f1oen2g 7031	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
20	6, 18, 19	mp3an12 1368	. . . . . . . 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 1952	. . . . . 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 7041	. . . . 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 6823	. 2 ⊢ (⊤ → ≈ Er V)
30	29	mptru 1411	1 ⊢ ≈ Er V

Syntax hints:   ∧ wa 104   ↔ wb 105  ⊤wtru 1403  ∃wex 1545   ∈ wcel 2209  Vcvv 2821   class class class wbr 4125  ^◡ccnv 4768   ∘ ccom 4773  Rel wrel 4774  –1-1-onto→wf1o 5371   Er wer 6794   ≈ cen 7010

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-er 6797  df-en 7013

This theorem is referenced by:  ensymb  7057  entr  7061