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Theorem fcoinver 29756
Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 29757. (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 5776 . . 3 Rel (𝐹𝐹)
21a1i 11 . 2 (𝐹 Fn 𝑋 → Rel (𝐹𝐹))
3 dmco 5786 . . 3 dom (𝐹𝐹) = (𝐹 “ dom 𝐹)
4 df-rn 5261 . . . . 5 ran 𝐹 = dom 𝐹
54imaeq2i 5604 . . . 4 (𝐹 “ ran 𝐹) = (𝐹 “ dom 𝐹)
6 cnvimarndm 5626 . . . . 5 (𝐹 “ ran 𝐹) = dom 𝐹
7 fndm 6129 . . . . 5 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
86, 7syl5eq 2817 . . . 4 (𝐹 Fn 𝑋 → (𝐹 “ ran 𝐹) = 𝑋)
95, 8syl5eqr 2819 . . 3 (𝐹 Fn 𝑋 → (𝐹 “ dom 𝐹) = 𝑋)
103, 9syl5eq 2817 . 2 (𝐹 Fn 𝑋 → dom (𝐹𝐹) = 𝑋)
11 cnvco 5445 . . . . 5 (𝐹𝐹) = (𝐹𝐹)
12 cnvcnvss 5729 . . . . . 6 𝐹𝐹
13 coss2 5416 . . . . . 6 (𝐹𝐹 → (𝐹𝐹) ⊆ (𝐹𝐹))
1412, 13ax-mp 5 . . . . 5 (𝐹𝐹) ⊆ (𝐹𝐹)
1511, 14eqsstri 3784 . . . 4 (𝐹𝐹) ⊆ (𝐹𝐹)
1615a1i 11 . . 3 (𝐹 Fn 𝑋(𝐹𝐹) ⊆ (𝐹𝐹))
17 coass 5797 . . . . 5 ((𝐹𝐹) ∘ (𝐹𝐹)) = (𝐹 ∘ (𝐹 ∘ (𝐹𝐹)))
18 coass 5797 . . . . . . 7 ((𝐹𝐹) ∘ 𝐹) = (𝐹 ∘ (𝐹𝐹))
19 fnfun 6127 . . . . . . . . . 10 (𝐹 Fn 𝑋 → Fun 𝐹)
20 funcocnv2 6303 . . . . . . . . . 10 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2119, 20syl 17 . . . . . . . . 9 (𝐹 Fn 𝑋 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2221coeq1d 5421 . . . . . . . 8 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ 𝐹) = (( I ↾ ran 𝐹) ∘ 𝐹))
23 dffn3 6195 . . . . . . . . 9 (𝐹 Fn 𝑋𝐹:𝑋⟶ran 𝐹)
24 fcoi2 6220 . . . . . . . . 9 (𝐹:𝑋⟶ran 𝐹 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
2523, 24sylbi 207 . . . . . . . 8 (𝐹 Fn 𝑋 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
2622, 25eqtrd 2805 . . . . . . 7 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ 𝐹) = 𝐹)
2718, 26syl5eqr 2819 . . . . . 6 (𝐹 Fn 𝑋 → (𝐹 ∘ (𝐹𝐹)) = 𝐹)
2827coeq2d 5422 . . . . 5 (𝐹 Fn 𝑋 → (𝐹 ∘ (𝐹 ∘ (𝐹𝐹))) = (𝐹𝐹))
2917, 28syl5eq 2817 . . . 4 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ (𝐹𝐹)) = (𝐹𝐹))
30 ssid 3773 . . . 4 (𝐹𝐹) ⊆ (𝐹𝐹)
3129, 30syl6eqss 3804 . . 3 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ (𝐹𝐹)) ⊆ (𝐹𝐹))
3216, 31unssd 3940 . 2 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∪ ((𝐹𝐹) ∘ (𝐹𝐹))) ⊆ (𝐹𝐹))
33 df-er 7900 . 2 ((𝐹𝐹) Er 𝑋 ↔ (Rel (𝐹𝐹) ∧ dom (𝐹𝐹) = 𝑋 ∧ ((𝐹𝐹) ∪ ((𝐹𝐹) ∘ (𝐹𝐹))) ⊆ (𝐹𝐹)))
342, 10, 32, 33syl3anbrc 1428 1 (𝐹 Fn 𝑋 → (𝐹𝐹) Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  cun 3721  wss 3723   I cid 5157  ccnv 5249  dom cdm 5250  ran crn 5251  cres 5252  cima 5253  ccom 5254  Rel wrel 5255  Fun wfun 6024   Fn wfn 6025  wf 6026   Er wer 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-fun 6032  df-fn 6033  df-f 6034  df-er 7900
This theorem is referenced by:  qtophaus  30243
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