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Theorem fcoinver 30374
Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 30375. (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 6068 . . 3 Rel (𝐹𝐹)
21a1i 11 . 2 (𝐹 Fn 𝑋 → Rel (𝐹𝐹))
3 dmco 6078 . . 3 dom (𝐹𝐹) = (𝐹 “ dom 𝐹)
4 df-rn 5534 . . . . 5 ran 𝐹 = dom 𝐹
54imaeq2i 5898 . . . 4 (𝐹 “ ran 𝐹) = (𝐹 “ dom 𝐹)
6 cnvimarndm 5921 . . . . 5 (𝐹 “ ran 𝐹) = dom 𝐹
7 fndm 6429 . . . . 5 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
86, 7syl5eq 2848 . . . 4 (𝐹 Fn 𝑋 → (𝐹 “ ran 𝐹) = 𝑋)
95, 8syl5eqr 2850 . . 3 (𝐹 Fn 𝑋 → (𝐹 “ dom 𝐹) = 𝑋)
103, 9syl5eq 2848 . 2 (𝐹 Fn 𝑋 → dom (𝐹𝐹) = 𝑋)
11 cnvco 5724 . . . . 5 (𝐹𝐹) = (𝐹𝐹)
12 cnvcnvss 6022 . . . . . 6 𝐹𝐹
13 coss2 5695 . . . . . 6 (𝐹𝐹 → (𝐹𝐹) ⊆ (𝐹𝐹))
1412, 13ax-mp 5 . . . . 5 (𝐹𝐹) ⊆ (𝐹𝐹)
1511, 14eqsstri 3952 . . . 4 (𝐹𝐹) ⊆ (𝐹𝐹)
1615a1i 11 . . 3 (𝐹 Fn 𝑋(𝐹𝐹) ⊆ (𝐹𝐹))
17 coass 6089 . . . . 5 ((𝐹𝐹) ∘ (𝐹𝐹)) = (𝐹 ∘ (𝐹 ∘ (𝐹𝐹)))
18 coass 6089 . . . . . . 7 ((𝐹𝐹) ∘ 𝐹) = (𝐹 ∘ (𝐹𝐹))
19 fnfun 6427 . . . . . . . . . 10 (𝐹 Fn 𝑋 → Fun 𝐹)
20 funcocnv2 6618 . . . . . . . . . 10 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2119, 20syl 17 . . . . . . . . 9 (𝐹 Fn 𝑋 → (𝐹𝐹) = ( I ↾ ran 𝐹))
2221coeq1d 5700 . . . . . . . 8 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ 𝐹) = (( I ↾ ran 𝐹) ∘ 𝐹))
23 dffn3 6503 . . . . . . . . 9 (𝐹 Fn 𝑋𝐹:𝑋⟶ran 𝐹)
24 fcoi2 6531 . . . . . . . . 9 (𝐹:𝑋⟶ran 𝐹 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
2523, 24sylbi 220 . . . . . . . 8 (𝐹 Fn 𝑋 → (( I ↾ ran 𝐹) ∘ 𝐹) = 𝐹)
2622, 25eqtrd 2836 . . . . . . 7 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ 𝐹) = 𝐹)
2718, 26syl5eqr 2850 . . . . . 6 (𝐹 Fn 𝑋 → (𝐹 ∘ (𝐹𝐹)) = 𝐹)
2827coeq2d 5701 . . . . 5 (𝐹 Fn 𝑋 → (𝐹 ∘ (𝐹 ∘ (𝐹𝐹))) = (𝐹𝐹))
2917, 28syl5eq 2848 . . . 4 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ (𝐹𝐹)) = (𝐹𝐹))
30 ssid 3940 . . . 4 (𝐹𝐹) ⊆ (𝐹𝐹)
3129, 30eqsstrdi 3972 . . 3 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∘ (𝐹𝐹)) ⊆ (𝐹𝐹))
3216, 31unssd 4116 . 2 (𝐹 Fn 𝑋 → ((𝐹𝐹) ∪ ((𝐹𝐹) ∘ (𝐹𝐹))) ⊆ (𝐹𝐹))
33 df-er 8276 . 2 ((𝐹𝐹) Er 𝑋 ↔ (Rel (𝐹𝐹) ∧ dom (𝐹𝐹) = 𝑋 ∧ ((𝐹𝐹) ∪ ((𝐹𝐹) ∘ (𝐹𝐹))) ⊆ (𝐹𝐹)))
342, 10, 32, 33syl3anbrc 1340 1 (𝐹 Fn 𝑋 → (𝐹𝐹) Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cun 3882  wss 3884   I cid 5427  ccnv 5522  dom cdm 5523  ran crn 5524  cres 5525  cima 5526  ccom 5527  Rel wrel 5528  Fun wfun 6322   Fn wfn 6323  wf 6324   Er wer 8273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-fun 6330  df-fn 6331  df-f 6332  df-er 8276
This theorem is referenced by:  qtophaus  31193
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