Theorem fcoinver 32584

Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 32585. (Contributed by Thierry Arnoux, 3-Jan-2020.)

Assertion

Ref	Expression
fcoinver	⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ 𝐹) Er 𝑋)

Proof of Theorem fcoinver

Step	Hyp	Ref	Expression
1		relco 6056	. . 3 ⊢ Rel (◡𝐹 ∘ 𝐹)
2	1	a1i 11	. 2 ⊢ (𝐹 Fn 𝑋 → Rel (◡𝐹 ∘ 𝐹))
3		dmco 6202	. . 3 ⊢ dom (◡𝐹 ∘ 𝐹) = (◡𝐹 “ dom ◡𝐹)
4		df-rn 5625	. . . . 5 ⊢ ran 𝐹 = dom ◡𝐹
5	4	imaeq2i 6006	. . . 4 ⊢ (◡𝐹 “ ran 𝐹) = (◡𝐹 “ dom ◡𝐹)
6		cnvimarndm 6031	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
7		fndm 6584	. . . . 5 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
8	6, 7	eqtrid 2778	. . . 4 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ ran 𝐹) = 𝑋)
9	5, 8	eqtr3id 2780	. . 3 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ dom ◡𝐹) = 𝑋)
10	3, 9	eqtrid 2778	. 2 ⊢ (𝐹 Fn 𝑋 → dom (◡𝐹 ∘ 𝐹) = 𝑋)
11		cnvco 5824	. . . . 5 ⊢ ◡(◡𝐹 ∘ 𝐹) = (◡𝐹 ∘ ◡◡𝐹)
12		cnvcnvss 6141	. . . . . 6 ⊢ ◡◡𝐹 ⊆ 𝐹
13		coss2 5795	. . . . . 6 ⊢ (◡◡𝐹 ⊆ 𝐹 → (◡𝐹 ∘ ◡◡𝐹) ⊆ (◡𝐹 ∘ 𝐹))
14	12, 13	ax-mp 5	. . . . 5 ⊢ (◡𝐹 ∘ ◡◡𝐹) ⊆ (◡𝐹 ∘ 𝐹)
15	11, 14	eqsstri 3976	. . . 4 ⊢ ◡(◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹)
16	15	a1i 11	. . 3 ⊢ (𝐹 Fn 𝑋 → ◡(◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹))
17		coass 6213	. . . . 5 ⊢ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐹 ∘ (𝐹 ∘ (◡𝐹 ∘ 𝐹)))
18		coass 6213	. . . . . . 7 ⊢ ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = (𝐹 ∘ (◡𝐹 ∘ 𝐹))
19		fnfun 6581	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
20		funcocnv2 6788	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
21	19, 20	syl 17	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
22	21	coeq1d 5800	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = (( I ↾ ran 𝐹) ∘ 𝐹))
23		dffn3 6663	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶ran 𝐹)
24		fcoi2 6698	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶ran 𝐹 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
25	23, 24	sylbi 217	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
26	22, 25	eqtrd 2766	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = 𝐹)
27	18, 26	eqtr3id 2780	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝐹 ∘ (◡𝐹 ∘ 𝐹)) = 𝐹)
28	27	coeq2d 5801	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ (𝐹 ∘ (◡𝐹 ∘ 𝐹))) = (◡𝐹 ∘ 𝐹))
29	17, 28	eqtrid 2778	. . . 4 ⊢ (𝐹 Fn 𝑋 → ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐹 ∘ 𝐹))
30		ssid 3952	. . . 4 ⊢ (◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹)
31	29, 30	eqsstrdi 3974	. . 3 ⊢ (𝐹 Fn 𝑋 → ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) ⊆ (◡𝐹 ∘ 𝐹))
32	16, 31	unssd 4139	. 2 ⊢ (𝐹 Fn 𝑋 → (◡(◡𝐹 ∘ 𝐹) ∪ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹))) ⊆ (◡𝐹 ∘ 𝐹))
33		df-er 8622	. 2 ⊢ ((◡𝐹 ∘ 𝐹) Er 𝑋 ↔ (Rel (◡𝐹 ∘ 𝐹) ∧ dom (◡𝐹 ∘ 𝐹) = 𝑋 ∧ (◡(◡𝐹 ∘ 𝐹) ∪ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹))) ⊆ (◡𝐹 ∘ 𝐹)))
34	2, 10, 32, 33	syl3anbrc 1344	1 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ 𝐹) Er 𝑋)

Syntax hints:   → wi 4   = wceq 1541   ∪ cun 3895   ⊆ wss 3897   I cid 5508  ^◡ccnv 5613  dom cdm 5614  ran crn 5615   ↾ cres 5616   “ cima 5617   ∘ ccom 5618  Rel wrel 5619  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477   Er wer 8619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-fun 6483  df-fn 6484  df-f 6485  df-er 8622

This theorem is referenced by:  qtophaus  33849