Theorem enerOLD 7889

Description: Obsolete proof of ener 7888 as of 1-May-2021. Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
enerOLD	⊢ ≈ Er V

Proof of Theorem enerOLD

Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relen 7846	. . . 4 ⊢ Rel ≈
2	1	a1i 11	. . 3 ⊢ (⊤ → Rel ≈ )
3		bren 7850	. . . . 5 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦)
4		f1ocnv 6062	. . . . . . 7 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → ◡𝑓:𝑦–1-1-onto→𝑥)
5		vex 3176	. . . . . . . 8 ⊢ 𝑦 ∈ V
6		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
7		f1oen2g 7858	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ ◡𝑓:𝑦–1-1-onto→𝑥) → 𝑦 ≈ 𝑥)
8	5, 6, 7	mp3an12 1406	. . . . . . 7 ⊢ (◡𝑓:𝑦–1-1-onto→𝑥 → 𝑦 ≈ 𝑥)
9	4, 8	syl 17	. . . . . 6 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥)
10	9	exlimiv 1845	. . . . 5 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → 𝑦 ≈ 𝑥)
11	3, 10	sylbi 206	. . . 4 ⊢ (𝑥 ≈ 𝑦 → 𝑦 ≈ 𝑥)
12	11	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ≈ 𝑦) → 𝑦 ≈ 𝑥)
13		bren 7850	. . . . 5 ⊢ (𝑥 ≈ 𝑦 ↔ ∃𝑔 𝑔:𝑥–1-1-onto→𝑦)
14		bren 7850	. . . . 5 ⊢ (𝑦 ≈ 𝑧 ↔ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧)
15		eeanv 2170	. . . . . 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 6072	. . . . . . . . 9 ⊢ ((𝑓:𝑦–1-1-onto→𝑧 ∧ 𝑔:𝑥–1-1-onto→𝑦) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧)
17	16	ancoms 468	. . . . . . . 8 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧)
18		vex 3176	. . . . . . . . 9 ⊢ 𝑧 ∈ V
19		f1oen2g 7858	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V ∧ (𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
20	6, 18, 19	mp3an12 1406	. . . . . . . 8 ⊢ ((𝑓 ∘ 𝑔):𝑥–1-1-onto→𝑧 → 𝑥 ≈ 𝑧)
21	17, 20	syl 17	. . . . . . 7 ⊢ ((𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
22	21	exlimivv 1847	. . . . . 6 ⊢ (∃𝑔∃𝑓(𝑔:𝑥–1-1-onto→𝑦 ∧ 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
23	15, 22	sylbir 224	. . . . 5 ⊢ ((∃𝑔 𝑔:𝑥–1-1-onto→𝑦 ∧ ∃𝑓 𝑓:𝑦–1-1-onto→𝑧) → 𝑥 ≈ 𝑧)
24	13, 14, 23	syl2anb 495	. . . 4 ⊢ ((𝑥 ≈ 𝑦 ∧ 𝑦 ≈ 𝑧) → 𝑥 ≈ 𝑧)
25	24	adantl 481	. . 3 ⊢ ((⊤ ∧ (𝑥 ≈ 𝑦 ∧ 𝑦 ≈ 𝑧)) → 𝑥 ≈ 𝑧)
26	6	enref 7874	. . . . 5 ⊢ 𝑥 ≈ 𝑥
27	6, 26	2th 253	. . . 4 ⊢ (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥)
28	27	a1i 11	. . 3 ⊢ (⊤ → (𝑥 ∈ V ↔ 𝑥 ≈ 𝑥))
29	2, 12, 25, 28	iserd 7655	. 2 ⊢ (⊤ → ≈ Er V)
30	29	trud 1484	1 ⊢ ≈ Er V

Syntax hints:   ↔ wb 195   ∧ wa 383  ⊤wtru 1476  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  ^◡ccnv 5037   ∘ ccom 5042  Rel wrel 5043  –1-1-onto→wf1o 5803   Er wer 7626   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-er 7629  df-en 7842

This theorem is referenced by: (None)