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Theorem fcoinver 29252
Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 29253. (Contributed by Thierry Arnoux, 3-Jan-2020.)
Assertion
Ref Expression
fcoinver (𝐹 Fn 𝑋 → (𝐹𝐹) Er 𝑋)

Proof of Theorem fcoinver
StepHypRef Expression
1 relco 5595 . . 3 Rel (𝐹𝐹)
21a1i 11 . 2 (𝐹 Fn 𝑋 → Rel (𝐹𝐹))
3 dmco 5605 . . 3 dom (𝐹𝐹) = (𝐹 “ dom 𝐹)
4 df-rn 5090 . . . . 5 ran 𝐹 = dom 𝐹
54imaeq2i 5427 . . . 4 (𝐹 “ ran 𝐹) = (𝐹 “ dom 𝐹)
6 cnvimarndm 5449 . . . . 5 (𝐹 “ ran 𝐹) = dom 𝐹
7 fndm 5950 . . . . 5 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
86, 7syl5eq 2672 . . . 4 (𝐹 Fn 𝑋 → (𝐹 “ ran 𝐹) = 𝑋)
95, 8syl5eqr 2674 . . 3 (𝐹 Fn 𝑋 → (𝐹 “ dom 𝐹) = 𝑋)
103, 9syl5eq 2672 . 2 (𝐹 Fn 𝑋 → dom (𝐹𝐹) = 𝑋)
11 cnvco 5273 . . . . 5 (𝐹𝐹) = (𝐹𝐹)
12 cnvcnvss 5551 . . . . . 6 𝐹𝐹
13 coss2 5243 . . . . . 6 (𝐹𝐹 → (𝐹𝐹) ⊆ (𝐹𝐹))
1412, 13ax-mp 5 . . . . 5 (𝐹𝐹) ⊆ (𝐹𝐹)
1511, 14eqsstri 3619 . . . 4 (𝐹𝐹) ⊆ (𝐹𝐹)
1615a1i 11 . . 3 (𝐹 Fn 𝑋(𝐹𝐹) ⊆ (𝐹𝐹))
17 coass 5616 . . . . 5 ((𝐹𝐹) ∘ (𝐹𝐹)) = (𝐹 ∘ (𝐹 ∘ (𝐹𝐹)))
18 coass 5616 . . . . . . 7 ((𝐹𝐹) ∘ 𝐹) = (𝐹 ∘ (𝐹𝐹))
19 fnfun 5948 . . . . . . . . . 10 (𝐹 Fn 𝑋 → Fun 𝐹)
20 funcocnv2 6120 . . . . . . . . . 10 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2119, 20syl 17 . . . . . . . . 9 (𝐹 Fn 𝑋 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2221coeq1d 5248 . . . . . . . 8 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ 𝐹) = (( I ↾ ran 𝐹) ∘ 𝐹))
23 dffn3 6013 . . . . . . . . 9 (𝐹 Fn 𝑋𝐹:𝑋⟶ran 𝐹)
24 fcoi2 6038 . . . . . . . . 9 (𝐹:𝑋⟶ran 𝐹 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
2523, 24sylbi 207 . . . . . . . 8 (𝐹 Fn 𝑋 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
2622, 25eqtrd 2660 . . . . . . 7 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ 𝐹) = 𝐹)
2718, 26syl5eqr 2674 . . . . . 6 (𝐹 Fn 𝑋 → (𝐹 ∘ (𝐹𝐹)) = 𝐹)
2827coeq2d 5249 . . . . 5 (𝐹 Fn 𝑋 → (𝐹 ∘ (𝐹 ∘ (𝐹𝐹))) = (𝐹𝐹))
2917, 28syl5eq 2672 . . . 4 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ (𝐹𝐹)) = (𝐹𝐹))
30 ssid 3608 . . . 4 (𝐹𝐹) ⊆ (𝐹𝐹)
3129, 30syl6eqss 3639 . . 3 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ (𝐹𝐹)) ⊆ (𝐹𝐹))
3216, 31unssd 3772 . 2 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∪ ((𝐹𝐹) ∘ (𝐹𝐹))) ⊆ (𝐹𝐹))
33 df-er 7688 . 2 ((𝐹𝐹) Er 𝑋 ↔ (Rel (𝐹𝐹) ∧ dom (𝐹𝐹) = 𝑋 ∧ ((𝐹𝐹) ∪ ((𝐹𝐹) ∘ (𝐹𝐹))) ⊆ (𝐹𝐹)))
342, 10, 32, 33syl3anbrc 1244 1 (𝐹 Fn 𝑋 → (𝐹𝐹) Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  cun 3558  wss 3560   I cid 4989  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  ccom 5083  Rel wrel 5084  Fun wfun 5844   Fn wfn 5845  wf 5846   Er wer 7685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-fun 5852  df-fn 5853  df-f 5854  df-er 7688
This theorem is referenced by:  qtophaus  29677
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