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Theorem brco3f1o 44046
Description: Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)
Hypotheses
Ref Expression
brco3f1o.c (𝜑𝐶:𝑌1-1-onto𝑍)
brco3f1o.d (𝜑𝐷:𝑋1-1-onto𝑌)
brco3f1o.e (𝜑𝐸:𝑊1-1-onto𝑋)
brco3f1o.r (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
Assertion
Ref Expression
brco3f1o (𝜑 → ((𝐶𝐵)𝐶𝐵 ∧ (𝐷‘(𝐶𝐵))𝐷(𝐶𝐵) ∧ 𝐴𝐸(𝐷‘(𝐶𝐵))))

Proof of Theorem brco3f1o
StepHypRef Expression
1 brco3f1o.e . . . 4 (𝜑𝐸:𝑊1-1-onto𝑋)
2 f1ocnv 6860 . . . 4 (𝐸:𝑊1-1-onto𝑋𝐸:𝑋1-1-onto𝑊)
3 f1ofn 6849 . . . 4 (𝐸:𝑋1-1-onto𝑊𝐸 Fn 𝑋)
41, 2, 33syl 18 . . 3 (𝜑𝐸 Fn 𝑋)
5 brco3f1o.d . . . 4 (𝜑𝐷:𝑋1-1-onto𝑌)
6 f1ocnv 6860 . . . 4 (𝐷:𝑋1-1-onto𝑌𝐷:𝑌1-1-onto𝑋)
7 f1of 6848 . . . 4 (𝐷:𝑌1-1-onto𝑋𝐷:𝑌𝑋)
85, 6, 73syl 18 . . 3 (𝜑𝐷:𝑌𝑋)
9 brco3f1o.c . . . 4 (𝜑𝐶:𝑌1-1-onto𝑍)
10 f1ocnv 6860 . . . 4 (𝐶:𝑌1-1-onto𝑍𝐶:𝑍1-1-onto𝑌)
11 f1of 6848 . . . 4 (𝐶:𝑍1-1-onto𝑌𝐶:𝑍𝑌)
129, 10, 113syl 18 . . 3 (𝜑𝐶:𝑍𝑌)
13 brco3f1o.r . . . 4 (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
14 relco 6126 . . . . . 6 Rel ((𝐶𝐷) ∘ 𝐸)
1514relbrcnv 6125 . . . . 5 (𝐵((𝐶𝐷) ∘ 𝐸)𝐴𝐴((𝐶𝐷) ∘ 𝐸)𝐵)
16 cnvco 5896 . . . . . . 7 ((𝐶𝐷) ∘ 𝐸) = (𝐸(𝐶𝐷))
17 cnvco 5896 . . . . . . . 8 (𝐶𝐷) = (𝐷𝐶)
1817coeq2i 5871 . . . . . . 7 (𝐸(𝐶𝐷)) = (𝐸 ∘ (𝐷𝐶))
1916, 18eqtri 2765 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐸 ∘ (𝐷𝐶))
2019breqi 5149 . . . . 5 (𝐵((𝐶𝐷) ∘ 𝐸)𝐴𝐵(𝐸 ∘ (𝐷𝐶))𝐴)
21 coass 6285 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷𝐸))
2221breqi 5149 . . . . 5 (𝐴((𝐶𝐷) ∘ 𝐸)𝐵𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
2315, 20, 223bitr3ri 302 . . . 4 (𝐴(𝐶 ∘ (𝐷𝐸))𝐵𝐵(𝐸 ∘ (𝐷𝐶))𝐴)
2413, 23sylib 218 . . 3 (𝜑𝐵(𝐸 ∘ (𝐷𝐶))𝐴)
254, 8, 12, 24brcofffn 44044 . 2 (𝜑 → (𝐵𝐶(𝐶𝐵) ∧ (𝐶𝐵)𝐷(𝐷‘(𝐶𝐵)) ∧ (𝐷‘(𝐶𝐵))𝐸𝐴))
26 f1orel 6851 . . . 4 (𝐶:𝑌1-1-onto𝑍 → Rel 𝐶)
27 relbrcnvg 6123 . . . 4 (Rel 𝐶 → (𝐵𝐶(𝐶𝐵) ↔ (𝐶𝐵)𝐶𝐵))
289, 26, 273syl 18 . . 3 (𝜑 → (𝐵𝐶(𝐶𝐵) ↔ (𝐶𝐵)𝐶𝐵))
29 f1orel 6851 . . . 4 (𝐷:𝑋1-1-onto𝑌 → Rel 𝐷)
30 relbrcnvg 6123 . . . 4 (Rel 𝐷 → ((𝐶𝐵)𝐷(𝐷‘(𝐶𝐵)) ↔ (𝐷‘(𝐶𝐵))𝐷(𝐶𝐵)))
315, 29, 303syl 18 . . 3 (𝜑 → ((𝐶𝐵)𝐷(𝐷‘(𝐶𝐵)) ↔ (𝐷‘(𝐶𝐵))𝐷(𝐶𝐵)))
32 f1orel 6851 . . . 4 (𝐸:𝑊1-1-onto𝑋 → Rel 𝐸)
33 relbrcnvg 6123 . . . 4 (Rel 𝐸 → ((𝐷‘(𝐶𝐵))𝐸𝐴𝐴𝐸(𝐷‘(𝐶𝐵))))
341, 32, 333syl 18 . . 3 (𝜑 → ((𝐷‘(𝐶𝐵))𝐸𝐴𝐴𝐸(𝐷‘(𝐶𝐵))))
3528, 31, 343anbi123d 1438 . 2 (𝜑 → ((𝐵𝐶(𝐶𝐵) ∧ (𝐶𝐵)𝐷(𝐷‘(𝐶𝐵)) ∧ (𝐷‘(𝐶𝐵))𝐸𝐴) ↔ ((𝐶𝐵)𝐶𝐵 ∧ (𝐷‘(𝐶𝐵))𝐷(𝐶𝐵) ∧ 𝐴𝐸(𝐷‘(𝐶𝐵)))))
3625, 35mpbid 232 1 (𝜑 → ((𝐶𝐵)𝐶𝐵 ∧ (𝐷‘(𝐶𝐵))𝐷(𝐶𝐵) ∧ 𝐴𝐸(𝐷‘(𝐶𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   class class class wbr 5143  ccnv 5684  ccom 5689  Rel wrel 5690   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by: (None)
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