Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brco3f1o Structured version   Visualization version   GIF version

Theorem brco3f1o 37799
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 6108 . . . 4 (𝐸:𝑊1-1-onto𝑋𝐸:𝑋1-1-onto𝑊)
3 f1ofn 6097 . . . 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 6108 . . . 4 (𝐷:𝑋1-1-onto𝑌𝐷:𝑌1-1-onto𝑋)
7 f1of 6096 . . . 4 (𝐷:𝑌1-1-onto𝑋𝐷:𝑌𝑋)
85, 6, 73syl 18 . . 3 (𝜑𝐷:𝑌𝑋)
9 brco3f1o.c . . . 4 (𝜑𝐶:𝑌1-1-onto𝑍)
10 f1ocnv 6108 . . . 4 (𝐶:𝑌1-1-onto𝑍𝐶:𝑍1-1-onto𝑌)
11 f1of 6096 . . . 4 (𝐶:𝑍1-1-onto𝑌𝐶:𝑍𝑌)
129, 10, 113syl 18 . . 3 (𝜑𝐶:𝑍𝑌)
13 brco3f1o.r . . . 4 (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
14 relco 5595 . . . . . 6 Rel ((𝐶𝐷) ∘ 𝐸)
1514relbrcnv 5469 . . . . 5 (𝐵((𝐶𝐷) ∘ 𝐸)𝐴𝐴((𝐶𝐷) ∘ 𝐸)𝐵)
16 cnvco 5273 . . . . . . 7 ((𝐶𝐷) ∘ 𝐸) = (𝐸(𝐶𝐷))
17 cnvco 5273 . . . . . . . 8 (𝐶𝐷) = (𝐷𝐶)
1817coeq2i 5247 . . . . . . 7 (𝐸(𝐶𝐷)) = (𝐸 ∘ (𝐷𝐶))
1916, 18eqtri 2648 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐸 ∘ (𝐷𝐶))
2019breqi 4624 . . . . 5 (𝐵((𝐶𝐷) ∘ 𝐸)𝐴𝐵(𝐸 ∘ (𝐷𝐶))𝐴)
21 coass 5616 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷𝐸))
2221breqi 4624 . . . . 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 37797 . 2 (𝜑 → (𝐵𝐶(𝐶𝐵) ∧ (𝐶𝐵)𝐷(𝐷‘(𝐶𝐵)) ∧ (𝐷‘(𝐶𝐵))𝐸𝐴))
26 f1orel 6099 . . . 4 (𝐶:𝑌1-1-onto𝑍 → Rel 𝐶)
27 relbrcnvg 5467 . . . 4 (Rel 𝐶 → (𝐵𝐶(𝐶𝐵) ↔ (𝐶𝐵)𝐶𝐵))
289, 26, 273syl 18 . . 3 (𝜑 → (𝐵𝐶(𝐶𝐵) ↔ (𝐶𝐵)𝐶𝐵))
29 f1orel 6099 . . . 4 (𝐷:𝑋1-1-onto𝑌 → Rel 𝐷)
30 relbrcnvg 5467 . . . 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 6099 . . . 4 (𝐸:𝑊1-1-onto𝑋 → Rel 𝐸)
33 relbrcnvg 5467 . . . 4 (Rel 𝐸 → ((𝐷‘(𝐶𝐵))𝐸𝐴𝐴𝐸(𝐷‘(𝐶𝐵))))
341, 32, 333syl 18 . . 3 (𝜑 → ((𝐷‘(𝐶𝐵))𝐸𝐴𝐴𝐸(𝐷‘(𝐶𝐵))))
3528, 31, 343anbi123d 1396 . 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 1036   class class class wbr 4618  ccnv 5078  ccom 5083  Rel wrel 5084   Fn wfn 5845  wf 5846  1-1-ontowf1o 5849  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator