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 43086
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 6844 . . . 4 (𝐸:𝑊1-1-onto𝑋𝐸:𝑋1-1-onto𝑊)
3 f1ofn 6833 . . . 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 6844 . . . 4 (𝐷:𝑋1-1-onto𝑌𝐷:𝑌1-1-onto𝑋)
7 f1of 6832 . . . 4 (𝐷:𝑌1-1-onto𝑋𝐷:𝑌𝑋)
85, 6, 73syl 18 . . 3 (𝜑𝐷:𝑌𝑋)
9 brco3f1o.c . . . 4 (𝜑𝐶:𝑌1-1-onto𝑍)
10 f1ocnv 6844 . . . 4 (𝐶:𝑌1-1-onto𝑍𝐶:𝑍1-1-onto𝑌)
11 f1of 6832 . . . 4 (𝐶:𝑍1-1-onto𝑌𝐶:𝑍𝑌)
129, 10, 113syl 18 . . 3 (𝜑𝐶:𝑍𝑌)
13 brco3f1o.r . . . 4 (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
14 relco 6106 . . . . . 6 Rel ((𝐶𝐷) ∘ 𝐸)
1514relbrcnv 6105 . . . . 5 (𝐵((𝐶𝐷) ∘ 𝐸)𝐴𝐴((𝐶𝐷) ∘ 𝐸)𝐵)
16 cnvco 5884 . . . . . . 7 ((𝐶𝐷) ∘ 𝐸) = (𝐸(𝐶𝐷))
17 cnvco 5884 . . . . . . . 8 (𝐶𝐷) = (𝐷𝐶)
1817coeq2i 5859 . . . . . . 7 (𝐸(𝐶𝐷)) = (𝐸 ∘ (𝐷𝐶))
1916, 18eqtri 2758 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐸 ∘ (𝐷𝐶))
2019breqi 5153 . . . . 5 (𝐵((𝐶𝐷) ∘ 𝐸)𝐴𝐵(𝐸 ∘ (𝐷𝐶))𝐴)
21 coass 6263 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷𝐸))
2221breqi 5153 . . . . 5 (𝐴((𝐶𝐷) ∘ 𝐸)𝐵𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
2315, 20, 223bitr3ri 301 . . . 4 (𝐴(𝐶 ∘ (𝐷𝐸))𝐵𝐵(𝐸 ∘ (𝐷𝐶))𝐴)
2413, 23sylib 217 . . 3 (𝜑𝐵(𝐸 ∘ (𝐷𝐶))𝐴)
254, 8, 12, 24brcofffn 43084 . 2 (𝜑 → (𝐵𝐶(𝐶𝐵) ∧ (𝐶𝐵)𝐷(𝐷‘(𝐶𝐵)) ∧ (𝐷‘(𝐶𝐵))𝐸𝐴))
26 f1orel 6835 . . . 4 (𝐶:𝑌1-1-onto𝑍 → Rel 𝐶)
27 relbrcnvg 6103 . . . 4 (Rel 𝐶 → (𝐵𝐶(𝐶𝐵) ↔ (𝐶𝐵)𝐶𝐵))
289, 26, 273syl 18 . . 3 (𝜑 → (𝐵𝐶(𝐶𝐵) ↔ (𝐶𝐵)𝐶𝐵))
29 f1orel 6835 . . . 4 (𝐷:𝑋1-1-onto𝑌 → Rel 𝐷)
30 relbrcnvg 6103 . . . 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 6835 . . . 4 (𝐸:𝑊1-1-onto𝑋 → Rel 𝐸)
33 relbrcnvg 6103 . . . 4 (Rel 𝐸 → ((𝐷‘(𝐶𝐵))𝐸𝐴𝐴𝐸(𝐷‘(𝐶𝐵))))
341, 32, 333syl 18 . . 3 (𝜑 → ((𝐷‘(𝐶𝐵))𝐸𝐴𝐴𝐸(𝐷‘(𝐶𝐵))))
3528, 31, 343anbi123d 1434 . 2 (𝜑 → ((𝐵𝐶(𝐶𝐵) ∧ (𝐶𝐵)𝐷(𝐷‘(𝐶𝐵)) ∧ (𝐷‘(𝐶𝐵))𝐸𝐴) ↔ ((𝐶𝐵)𝐶𝐵 ∧ (𝐷‘(𝐶𝐵))𝐷(𝐶𝐵) ∧ 𝐴𝐸(𝐷‘(𝐶𝐵)))))
3625, 35mpbid 231 1 (𝜑 → ((𝐶𝐵)𝐶𝐵 ∧ (𝐷‘(𝐶𝐵))𝐷(𝐶𝐵) ∧ 𝐴𝐸(𝐷‘(𝐶𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1085   class class class wbr 5147  ccnv 5674  ccom 5679  Rel wrel 5680   Fn wfn 6537  wf 6538  1-1-ontowf1o 6541  cfv 6542
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator