Theorem fcoinver 31054

Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 31055. (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 6031	. . 3 ⊢ Rel (◡𝐹 ∘ 𝐹)
2	1	a1i 11	. 2 ⊢ (𝐹 Fn 𝑋 → Rel (◡𝐹 ∘ 𝐹))
3		dmco 6177	. . 3 ⊢ dom (◡𝐹 ∘ 𝐹) = (◡𝐹 “ dom ◡𝐹)
4		df-rn 5616	. . . . 5 ⊢ ran 𝐹 = dom ◡𝐹
5	4	imaeq2i 5982	. . . 4 ⊢ (◡𝐹 “ ran 𝐹) = (◡𝐹 “ dom ◡𝐹)
6		cnvimarndm 6005	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
7		fndm 6572	. . . . 5 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
8	6, 7	eqtrid 2789	. . . 4 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ ran 𝐹) = 𝑋)
9	5, 8	eqtr3id 2791	. . 3 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ dom ◡𝐹) = 𝑋)
10	3, 9	eqtrid 2789	. 2 ⊢ (𝐹 Fn 𝑋 → dom (◡𝐹 ∘ 𝐹) = 𝑋)
11		cnvco 5812	. . . . 5 ⊢ ◡(◡𝐹 ∘ 𝐹) = (◡𝐹 ∘ ◡◡𝐹)
12		cnvcnvss 6117	. . . . . 6 ⊢ ◡◡𝐹 ⊆ 𝐹
13		coss2 5783	. . . . . 6 ⊢ (◡◡𝐹 ⊆ 𝐹 → (◡𝐹 ∘ ◡◡𝐹) ⊆ (◡𝐹 ∘ 𝐹))
14	12, 13	ax-mp 5	. . . . 5 ⊢ (◡𝐹 ∘ ◡◡𝐹) ⊆ (◡𝐹 ∘ 𝐹)
15	11, 14	eqsstri 3964	. . . 4 ⊢ ◡(◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹)
16	15	a1i 11	. . 3 ⊢ (𝐹 Fn 𝑋 → ◡(◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹))
17		coass 6188	. . . . 5 ⊢ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐹 ∘ (𝐹 ∘ (◡𝐹 ∘ 𝐹)))
18		coass 6188	. . . . . . 7 ⊢ ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = (𝐹 ∘ (◡𝐹 ∘ 𝐹))
19		fnfun 6569	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
20		funcocnv2 6776	. . . . . . . . . 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 5788	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = (( I ↾ ran 𝐹) ∘ 𝐹))
23		dffn3 6648	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶ran 𝐹)
24		fcoi2 6684	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶ran 𝐹 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
25	23, 24	sylbi 216	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
26	22, 25	eqtrd 2777	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → ((𝐹 ∘ ◡𝐹) ∘ 𝐹) = 𝐹)
27	18, 26	eqtr3id 2791	. . . . . 6 ⊢ (𝐹 Fn 𝑋 → (𝐹 ∘ (◡𝐹 ∘ 𝐹)) = 𝐹)
28	27	coeq2d 5789	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ (𝐹 ∘ (◡𝐹 ∘ 𝐹))) = (◡𝐹 ∘ 𝐹))
29	17, 28	eqtrid 2789	. . . 4 ⊢ (𝐹 Fn 𝑋 → ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐹 ∘ 𝐹))
30		ssid 3952	. . . 4 ⊢ (◡𝐹 ∘ 𝐹) ⊆ (◡𝐹 ∘ 𝐹)
31	29, 30	eqsstrdi 3984	. . 3 ⊢ (𝐹 Fn 𝑋 → ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹)) ⊆ (◡𝐹 ∘ 𝐹))
32	16, 31	unssd 4130	. 2 ⊢ (𝐹 Fn 𝑋 → (◡(◡𝐹 ∘ 𝐹) ∪ ((◡𝐹 ∘ 𝐹) ∘ (◡𝐹 ∘ 𝐹))) ⊆ (◡𝐹 ∘ 𝐹))
33		df-er 8544	. 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 1342	1 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 ∘ 𝐹) Er 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∪ cun 3894   ⊆ wss 3896   I cid 5504  ^◡ccnv 5604  dom cdm 5605  ran crn 5606   ↾ cres 5607   “ cima 5608   ∘ ccom 5609  Rel wrel 5610  Fun wfun 6457   Fn wfn 6458  ⟶wf 6459   Er wer 8541

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pr 5365

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5086  df-opab 5148  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-fun 6465  df-fn 6466  df-f 6467  df-er 8544

This theorem is referenced by:  qtophaus  31892