Theorem fcoinver 32583

Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 32584. (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 6068	. . 3 ⊢ Rel (◡𝐹 ∘ 𝐹)
2	1	a1i 11	. 2 ⊢ (𝐹 Fn 𝑋 → Rel (◡𝐹 ∘ 𝐹))
3		dmco 6215	. . 3 ⊢ dom (◡𝐹 ∘ 𝐹) = (◡𝐹 “ dom ◡𝐹)
4		df-rn 5642	. . . . 5 ⊢ ran 𝐹 = dom ◡𝐹
5	4	imaeq2i 6018	. . . 4 ⊢ (◡𝐹 “ ran 𝐹) = (◡𝐹 “ dom ◡𝐹)
6		cnvimarndm 6043	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
7		fndm 6603	. . . . 5 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
8	6, 7	eqtrid 2776	. . . 4 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ ran 𝐹) = 𝑋)
9	5, 8	eqtr3id 2778	. . 3 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ dom ◡𝐹) = 𝑋)
10	3, 9	eqtrid 2776	. 2 ⊢ (𝐹 Fn 𝑋 → dom (◡𝐹 ∘ 𝐹) = 𝑋)
11		cnvco 5839	. . . . 5 ⊢ ◡(◡𝐹 ∘ 𝐹) = (◡𝐹 ∘ ◡◡𝐹)
12		cnvcnvss 6155	. . . . . 6 ⊢ ◡◡𝐹 ⊆ 𝐹
13		coss2 5810	. . . . . 6 ⊢ (◡◡𝐹 ⊆ 𝐹 → (◡𝐹 ∘ ◡◡𝐹) ⊆ (◡𝐹 ∘ 𝐹))
14	12, 13	ax-mp 5	. . . . 5 ⊢ (◡𝐹 ∘ ◡◡𝐹) ⊆ (◡𝐹 ∘ 𝐹)
15	11, 14	eqsstri 3990	. . . 4 ⊢ ◡(◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹)
16	15	a1i 11	. . 3 ⊢ (𝐹 Fn 𝑋 → ◡(◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹))
17		coass 6226	. . . . 5 ⊢ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐹 ∘ (𝐹 ∘ (◡𝐹 ∘ 𝐹)))
18		coass 6226	. . . . . . 7 ⊢ ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = (𝐹 ∘ (◡𝐹 ∘ 𝐹))
19		fnfun 6600	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
20		funcocnv2 6807	. . . . . . . . . 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 5815	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = (( I ↾ ran 𝐹) ∘ 𝐹))
23		dffn3 6682	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶ran 𝐹)
24		fcoi2 6717	. . . . . . . . 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 2764	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = 𝐹)
27	18, 26	eqtr3id 2778	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝐹 ∘ (◡𝐹 ∘ 𝐹)) = 𝐹)
28	27	coeq2d 5816	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ (𝐹 ∘ (◡𝐹 ∘ 𝐹))) = (◡𝐹 ∘ 𝐹))
29	17, 28	eqtrid 2776	. . . 4 ⊢ (𝐹 Fn 𝑋 → ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐹 ∘ 𝐹))
30		ssid 3966	. . . 4 ⊢ (◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹)
31	29, 30	eqsstrdi 3988	. . 3 ⊢ (𝐹 Fn 𝑋 → ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) ⊆ (◡𝐹 ∘ 𝐹))
32	16, 31	unssd 4151	. 2 ⊢ (𝐹 Fn 𝑋 → (◡(◡𝐹 ∘ 𝐹) ∪ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹))) ⊆ (◡𝐹 ∘ 𝐹))
33		df-er 8648	. 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 1540   ∪ cun 3909   ⊆ wss 3911   I cid 5525  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634   ∘ ccom 5635  Rel wrel 5636  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495   Er wer 8645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6501  df-fn 6502  df-f 6503  df-er 8648

This theorem is referenced by:  qtophaus  33819