Theorem fcoinver 30619

Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 30620. (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 6088	. . 3 ⊢ Rel (◡𝐹 ∘ 𝐹)
2	1	a1i 11	. 2 ⊢ (𝐹 Fn 𝑋 → Rel (◡𝐹 ∘ 𝐹))
3		dmco 6098	. . 3 ⊢ dom (◡𝐹 ∘ 𝐹) = (◡𝐹 “ dom ◡𝐹)
4		df-rn 5547	. . . . 5 ⊢ ran 𝐹 = dom ◡𝐹
5	4	imaeq2i 5912	. . . 4 ⊢ (◡𝐹 “ ran 𝐹) = (◡𝐹 “ dom ◡𝐹)
6		cnvimarndm 5935	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
7		fndm 6459	. . . . 5 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
8	6, 7	syl5eq 2783	. . . 4 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ ran 𝐹) = 𝑋)
9	5, 8	eqtr3id 2785	. . 3 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ dom ◡𝐹) = 𝑋)
10	3, 9	syl5eq 2783	. 2 ⊢ (𝐹 Fn 𝑋 → dom (◡𝐹 ∘ 𝐹) = 𝑋)
11		cnvco 5739	. . . . 5 ⊢ ◡(◡𝐹 ∘ 𝐹) = (◡𝐹 ∘ ◡◡𝐹)
12		cnvcnvss 6037	. . . . . 6 ⊢ ◡◡𝐹 ⊆ 𝐹
13		coss2 5710	. . . . . 6 ⊢ (◡◡𝐹 ⊆ 𝐹 → (◡𝐹 ∘ ◡◡𝐹) ⊆ (◡𝐹 ∘ 𝐹))
14	12, 13	ax-mp 5	. . . . 5 ⊢ (◡𝐹 ∘ ◡◡𝐹) ⊆ (◡𝐹 ∘ 𝐹)
15	11, 14	eqsstri 3921	. . . 4 ⊢ ◡(◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹)
16	15	a1i 11	. . 3 ⊢ (𝐹 Fn 𝑋 → ◡(◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹))
17		coass 6109	. . . . 5 ⊢ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐹 ∘ (𝐹 ∘ (◡𝐹 ∘ 𝐹)))
18		coass 6109	. . . . . . 7 ⊢ ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = (𝐹 ∘ (◡𝐹 ∘ 𝐹))
19		fnfun 6457	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
20		funcocnv2 6663	. . . . . . . . . 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 5715	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = (( I ↾ ran 𝐹) ∘ 𝐹))
23		dffn3 6536	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶ran 𝐹)
24		fcoi2 6572	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶ran 𝐹 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
25	23, 24	sylbi 220	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
26	22, 25	eqtrd 2771	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = 𝐹)
27	18, 26	eqtr3id 2785	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝐹 ∘ (◡𝐹 ∘ 𝐹)) = 𝐹)
28	27	coeq2d 5716	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ (𝐹 ∘ (◡𝐹 ∘ 𝐹))) = (◡𝐹 ∘ 𝐹))
29	17, 28	syl5eq 2783	. . . 4 ⊢ (𝐹 Fn 𝑋 → ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐹 ∘ 𝐹))
30		ssid 3909	. . . 4 ⊢ (◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹)
31	29, 30	eqsstrdi 3941	. . 3 ⊢ (𝐹 Fn 𝑋 → ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) ⊆ (◡𝐹 ∘ 𝐹))
32	16, 31	unssd 4086	. 2 ⊢ (𝐹 Fn 𝑋 → (◡(◡𝐹 ∘ 𝐹) ∪ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹))) ⊆ (◡𝐹 ∘ 𝐹))
33		df-er 8369	. 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 1345	1 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ 𝐹) Er 𝑋)

Syntax hints:   → wi 4   = wceq 1543   ∪ cun 3851   ⊆ wss 3853   I cid 5439  ^◡ccnv 5535  dom cdm 5536  ran crn 5537   ↾ cres 5538   “ cima 5539   ∘ ccom 5540  Rel wrel 5541  Fun wfun 6352   Fn wfn 6353  ⟶wf 6354   Er wer 8366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-fun 6360  df-fn 6361  df-f 6362  df-er 8369

This theorem is referenced by:  qtophaus  31454