Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fcoinver Structured version   Visualization version   GIF version

Theorem fcoinver 30047
Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 30048. (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 5972 . . 3 Rel (𝐹𝐹)
21a1i 11 . 2 (𝐹 Fn 𝑋 → Rel (𝐹𝐹))
3 dmco 5982 . . 3 dom (𝐹𝐹) = (𝐹 “ dom 𝐹)
4 df-rn 5454 . . . . 5 ran 𝐹 = dom 𝐹
54imaeq2i 5804 . . . 4 (𝐹 “ ran 𝐹) = (𝐹 “ dom 𝐹)
6 cnvimarndm 5826 . . . . 5 (𝐹 “ ran 𝐹) = dom 𝐹
7 fndm 6325 . . . . 5 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
86, 7syl5eq 2843 . . . 4 (𝐹 Fn 𝑋 → (𝐹 “ ran 𝐹) = 𝑋)
95, 8syl5eqr 2845 . . 3 (𝐹 Fn 𝑋 → (𝐹 “ dom 𝐹) = 𝑋)
103, 9syl5eq 2843 . 2 (𝐹 Fn 𝑋 → dom (𝐹𝐹) = 𝑋)
11 cnvco 5642 . . . . 5 (𝐹𝐹) = (𝐹𝐹)
12 cnvcnvss 5927 . . . . . 6 𝐹𝐹
13 coss2 5613 . . . . . 6 (𝐹𝐹 → (𝐹𝐹) ⊆ (𝐹𝐹))
1412, 13ax-mp 5 . . . . 5 (𝐹𝐹) ⊆ (𝐹𝐹)
1511, 14eqsstri 3922 . . . 4 (𝐹𝐹) ⊆ (𝐹𝐹)
1615a1i 11 . . 3 (𝐹 Fn 𝑋(𝐹𝐹) ⊆ (𝐹𝐹))
17 coass 5993 . . . . 5 ((𝐹𝐹) ∘ (𝐹𝐹)) = (𝐹 ∘ (𝐹 ∘ (𝐹𝐹)))
18 coass 5993 . . . . . . 7 ((𝐹𝐹) ∘ 𝐹) = (𝐹 ∘ (𝐹𝐹))
19 fnfun 6323 . . . . . . . . . 10 (𝐹 Fn 𝑋 → Fun 𝐹)
20 funcocnv2 6507 . . . . . . . . . 10 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2119, 20syl 17 . . . . . . . . 9 (𝐹 Fn 𝑋 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2221coeq1d 5618 . . . . . . . 8 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ 𝐹) = (( I ↾ ran 𝐹) ∘ 𝐹))
23 dffn3 6393 . . . . . . . . 9 (𝐹 Fn 𝑋𝐹:𝑋⟶ran 𝐹)
24 fcoi2 6421 . . . . . . . . 9 (𝐹:𝑋⟶ran 𝐹 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
2523, 24sylbi 218 . . . . . . . 8 (𝐹 Fn 𝑋 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
2622, 25eqtrd 2831 . . . . . . 7 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ 𝐹) = 𝐹)
2718, 26syl5eqr 2845 . . . . . 6 (𝐹 Fn 𝑋 → (𝐹 ∘ (𝐹𝐹)) = 𝐹)
2827coeq2d 5619 . . . . 5 (𝐹 Fn 𝑋 → (𝐹 ∘ (𝐹 ∘ (𝐹𝐹))) = (𝐹𝐹))
2917, 28syl5eq 2843 . . . 4 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ (𝐹𝐹)) = (𝐹𝐹))
30 ssid 3910 . . . 4 (𝐹𝐹) ⊆ (𝐹𝐹)
3129, 30syl6eqss 3942 . . 3 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ (𝐹𝐹)) ⊆ (𝐹𝐹))
3216, 31unssd 4083 . 2 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∪ ((𝐹𝐹) ∘ (𝐹𝐹))) ⊆ (𝐹𝐹))
33 df-er 8139 . 2 ((𝐹𝐹) Er 𝑋 ↔ (Rel (𝐹𝐹) ∧ dom (𝐹𝐹) = 𝑋 ∧ ((𝐹𝐹) ∪ ((𝐹𝐹) ∘ (𝐹𝐹))) ⊆ (𝐹𝐹)))
342, 10, 32, 33syl3anbrc 1336 1 (𝐹 Fn 𝑋 → (𝐹𝐹) Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1522  cun 3857  wss 3859   I cid 5347  ccnv 5442  dom cdm 5443  ran crn 5444  cres 5445  cima 5446  ccom 5447  Rel wrel 5448  Fun wfun 6219   Fn wfn 6220  wf 6221   Er wer 8136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-fun 6227  df-fn 6228  df-f 6229  df-er 8139
This theorem is referenced by:  qtophaus  30717
  Copyright terms: Public domain W3C validator