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Theorem brco3f1o 38648
 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 6187 . . . 4 (𝐸:𝑊1-1-onto𝑋𝐸:𝑋1-1-onto𝑊)
3 f1ofn 6176 . . . 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 6187 . . . 4 (𝐷:𝑋1-1-onto𝑌𝐷:𝑌1-1-onto𝑋)
7 f1of 6175 . . . 4 (𝐷:𝑌1-1-onto𝑋𝐷:𝑌𝑋)
85, 6, 73syl 18 . . 3 (𝜑𝐷:𝑌𝑋)
9 brco3f1o.c . . . 4 (𝜑𝐶:𝑌1-1-onto𝑍)
10 f1ocnv 6187 . . . 4 (𝐶:𝑌1-1-onto𝑍𝐶:𝑍1-1-onto𝑌)
11 f1of 6175 . . . 4 (𝐶:𝑍1-1-onto𝑌𝐶:𝑍𝑌)
129, 10, 113syl 18 . . 3 (𝜑𝐶:𝑍𝑌)
13 brco3f1o.r . . . 4 (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
14 relco 5671 . . . . . 6 Rel ((𝐶𝐷) ∘ 𝐸)
1514relbrcnv 5541 . . . . 5 (𝐵((𝐶𝐷) ∘ 𝐸)𝐴𝐴((𝐶𝐷) ∘ 𝐸)𝐵)
16 cnvco 5340 . . . . . . 7 ((𝐶𝐷) ∘ 𝐸) = (𝐸(𝐶𝐷))
17 cnvco 5340 . . . . . . . 8 (𝐶𝐷) = (𝐷𝐶)
1817coeq2i 5315 . . . . . . 7 (𝐸(𝐶𝐷)) = (𝐸 ∘ (𝐷𝐶))
1916, 18eqtri 2673 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐸 ∘ (𝐷𝐶))
2019breqi 4691 . . . . 5 (𝐵((𝐶𝐷) ∘ 𝐸)𝐴𝐵(𝐸 ∘ (𝐷𝐶))𝐴)
21 coass 5692 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷𝐸))
2221breqi 4691 . . . . 5 (𝐴((𝐶𝐷) ∘ 𝐸)𝐵𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
2315, 20, 223bitr3ri 291 . . . 4 (𝐴(𝐶 ∘ (𝐷𝐸))𝐵𝐵(𝐸 ∘ (𝐷𝐶))𝐴)
2413, 23sylib 208 . . 3 (𝜑𝐵(𝐸 ∘ (𝐷𝐶))𝐴)
254, 8, 12, 24brcofffn 38646 . 2 (𝜑 → (𝐵𝐶(𝐶𝐵) ∧ (𝐶𝐵)𝐷(𝐷‘(𝐶𝐵)) ∧ (𝐷‘(𝐶𝐵))𝐸𝐴))
26 f1orel 6178 . . . 4 (𝐶:𝑌1-1-onto𝑍 → Rel 𝐶)
27 relbrcnvg 5539 . . . 4 (Rel 𝐶 → (𝐵𝐶(𝐶𝐵) ↔ (𝐶𝐵)𝐶𝐵))
289, 26, 273syl 18 . . 3 (𝜑 → (𝐵𝐶(𝐶𝐵) ↔ (𝐶𝐵)𝐶𝐵))
29 f1orel 6178 . . . 4 (𝐷:𝑋1-1-onto𝑌 → Rel 𝐷)
30 relbrcnvg 5539 . . . 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 6178 . . . 4 (𝐸:𝑊1-1-onto𝑋 → Rel 𝐸)
33 relbrcnvg 5539 . . . 4 (Rel 𝐸 → ((𝐷‘(𝐶𝐵))𝐸𝐴𝐴𝐸(𝐷‘(𝐶𝐵))))
341, 32, 333syl 18 . . 3 (𝜑 → ((𝐷‘(𝐶𝐵))𝐸𝐴𝐴𝐸(𝐷‘(𝐶𝐵))))
3528, 31, 343anbi123d 1439 . 2 (𝜑 → ((𝐵𝐶(𝐶𝐵) ∧ (𝐶𝐵)𝐷(𝐷‘(𝐶𝐵)) ∧ (𝐷‘(𝐶𝐵))𝐸𝐴) ↔ ((𝐶𝐵)𝐶𝐵 ∧ (𝐷‘(𝐶𝐵))𝐷(𝐶𝐵) ∧ 𝐴𝐸(𝐷‘(𝐶𝐵)))))
3625, 35mpbid 222 1 (𝜑 → ((𝐶𝐵)𝐶𝐵 ∧ (𝐷‘(𝐶𝐵))𝐷(𝐶𝐵) ∧ 𝐴𝐸(𝐷‘(𝐶𝐵))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1054   class class class wbr 4685  ◡ccnv 5142   ∘ ccom 5147  Rel wrel 5148   Fn wfn 5921  ⟶wf 5922  –1-1-onto→wf1o 5925  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934 This theorem is referenced by: (None)
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