Theorem fcoinver 28798

Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 28799. (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 5550	. . 3 ⊢ Rel (◡𝐹 ∘ 𝐹)
2	1	a1i 11	. 2 ⊢ (𝐹 Fn 𝑋 → Rel (◡𝐹 ∘ 𝐹))
3		dmco 5560	. . 3 ⊢ dom (◡𝐹 ∘ 𝐹) = (◡𝐹 “ dom ◡𝐹)
4		df-rn 5049	. . . . 5 ⊢ ran 𝐹 = dom ◡𝐹
5	4	imaeq2i 5383	. . . 4 ⊢ (◡𝐹 “ ran 𝐹) = (◡𝐹 “ dom ◡𝐹)
6		cnvimarndm 5405	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
7		fndm 5904	. . . . 5 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
8	6, 7	syl5eq 2656	. . . 4 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ ran 𝐹) = 𝑋)
9	5, 8	syl5eqr 2658	. . 3 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ dom ◡𝐹) = 𝑋)
10	3, 9	syl5eq 2656	. 2 ⊢ (𝐹 Fn 𝑋 → dom (◡𝐹 ∘ 𝐹) = 𝑋)
11		cnvco 5230	. . . . 5 ⊢ ◡(◡𝐹 ∘ 𝐹) = (◡𝐹 ∘ ◡◡𝐹)
12		cnvcnvss 5507	. . . . . 6 ⊢ ◡◡𝐹 ⊆ 𝐹
13		coss2 5200	. . . . . 6 ⊢ (◡◡𝐹 ⊆ 𝐹 → (◡𝐹 ∘ ◡◡𝐹) ⊆ (◡𝐹 ∘ 𝐹))
14	12, 13	ax-mp 5	. . . . 5 ⊢ (◡𝐹 ∘ ◡◡𝐹) ⊆ (◡𝐹 ∘ 𝐹)
15	11, 14	eqsstri 3598	. . . 4 ⊢ ◡(◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹)
16	15	a1i 11	. . 3 ⊢ (𝐹 Fn 𝑋 → ◡(◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹))
17		coass 5571	. . . . 5 ⊢ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐹 ∘ (𝐹 ∘ (◡𝐹 ∘ 𝐹)))
18		coass 5571	. . . . . . 7 ⊢ ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = (𝐹 ∘ (◡𝐹 ∘ 𝐹))
19		fnfun 5902	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
20		funcocnv2 6074	. . . . . . . . . 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 5205	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = (( I ↾ ran 𝐹) ∘ 𝐹))
23		dffn3 5967	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶ran 𝐹)
24		fcoi2 5992	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶ran 𝐹 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
25	23, 24	sylbi 206	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
26	22, 25	eqtrd 2644	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = 𝐹)
27	18, 26	syl5eqr 2658	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝐹 ∘ (◡𝐹 ∘ 𝐹)) = 𝐹)
28	27	coeq2d 5206	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ (𝐹 ∘ (◡𝐹 ∘ 𝐹))) = (◡𝐹 ∘ 𝐹))
29	17, 28	syl5eq 2656	. . . 4 ⊢ (𝐹 Fn 𝑋 → ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐹 ∘ 𝐹))
30		ssid 3587	. . . 4 ⊢ (◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹)
31	29, 30	syl6eqss 3618	. . 3 ⊢ (𝐹 Fn 𝑋 → ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) ⊆ (◡𝐹 ∘ 𝐹))
32	16, 31	unssd 3751	. 2 ⊢ (𝐹 Fn 𝑋 → (◡(◡𝐹 ∘ 𝐹) ∪ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹))) ⊆ (◡𝐹 ∘ 𝐹))
33		df-er 7629	. 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 1239	1 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ 𝐹) Er 𝑋)

Syntax hints:   → wi 4   = wceq 1475   ∪ cun 3538   ⊆ wss 3540   I cid 4948  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800   Er wer 7626

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

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-sn 4126  df-pr 4128  df-op 4132  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-er 7629

This theorem is referenced by:  qtophaus  29231