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

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 → (𝐹𝐹) Er 𝑋)

Proof of Theorem fcoinver
StepHypRef Expression
1 relco 5550 . . 3 Rel (𝐹𝐹)
21a1i 11 . 2 (𝐹 Fn 𝑋 → Rel (𝐹𝐹))
3 dmco 5560 . . 3 dom (𝐹𝐹) = (𝐹 “ dom 𝐹)
4 df-rn 5049 . . . . 5 ran 𝐹 = dom 𝐹
54imaeq2i 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)
86, 7syl5eq 2656 . . . 4 (𝐹 Fn 𝑋 → (𝐹 “ ran 𝐹) = 𝑋)
95, 8syl5eqr 2658 . . 3 (𝐹 Fn 𝑋 → (𝐹 “ dom 𝐹) = 𝑋)
103, 9syl5eq 2656 . 2 (𝐹 Fn 𝑋 → dom (𝐹𝐹) = 𝑋)
11 cnvco 5230 . . . . 5 (𝐹𝐹) = (𝐹𝐹)
12 cnvcnvss 5507 . . . . . 6 𝐹𝐹
13 coss2 5200 . . . . . 6 (𝐹𝐹 → (𝐹𝐹) ⊆ (𝐹𝐹))
1412, 13ax-mp 5 . . . . 5 (𝐹𝐹) ⊆ (𝐹𝐹)
1511, 14eqsstri 3598 . . . 4 (𝐹𝐹) ⊆ (𝐹𝐹)
1615a1i 11 . . 3 (𝐹 Fn 𝑋(𝐹𝐹) ⊆ (𝐹𝐹))
17 coass 5571 . . . . 5 ((𝐹𝐹) ∘ (𝐹𝐹)) = (𝐹 ∘ (𝐹 ∘ (𝐹𝐹)))
18 coass 5571 . . . . . . 7 ((𝐹𝐹) ∘ 𝐹) = (𝐹 ∘ (𝐹𝐹))
19 fnfun 5902 . . . . . . . . . 10 (𝐹 Fn 𝑋 → Fun 𝐹)
20 funcocnv2 6074 . . . . . . . . . 10 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2119, 20syl 17 . . . . . . . . 9 (𝐹 Fn 𝑋 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2221coeq1d 5205 . . . . . . . 8 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ 𝐹) = (( I ↾ ran 𝐹) ∘ 𝐹))
23 dffn3 5967 . . . . . . . . 9 (𝐹 Fn 𝑋𝐹:𝑋⟶ran 𝐹)
24 fcoi2 5992 . . . . . . . . 9 (𝐹:𝑋⟶ran 𝐹 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
2523, 24sylbi 206 . . . . . . . 8 (𝐹 Fn 𝑋 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
2622, 25eqtrd 2644 . . . . . . 7 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ 𝐹) = 𝐹)
2718, 26syl5eqr 2658 . . . . . 6 (𝐹 Fn 𝑋 → (𝐹 ∘ (𝐹𝐹)) = 𝐹)
2827coeq2d 5206 . . . . 5 (𝐹 Fn 𝑋 → (𝐹 ∘ (𝐹 ∘ (𝐹𝐹))) = (𝐹𝐹))
2917, 28syl5eq 2656 . . . 4 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ (𝐹𝐹)) = (𝐹𝐹))
30 ssid 3587 . . . 4 (𝐹𝐹) ⊆ (𝐹𝐹)
3129, 30syl6eqss 3618 . . 3 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ (𝐹𝐹)) ⊆ (𝐹𝐹))
3216, 31unssd 3751 . 2 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∪ ((𝐹𝐹) ∘ (𝐹𝐹))) ⊆ (𝐹𝐹))
33 df-er 7629 . 2 ((𝐹𝐹) Er 𝑋 ↔ (Rel (𝐹𝐹) ∧ dom (𝐹𝐹) = 𝑋 ∧ ((𝐹𝐹) ∪ ((𝐹𝐹) ∘ (𝐹𝐹))) ⊆ (𝐹𝐹)))
342, 10, 32, 33syl3anbrc 1239 1 (𝐹 Fn 𝑋 → (𝐹𝐹) Er 𝑋)
 Colors of variables: wff setvar class 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
 Copyright terms: Public domain W3C validator