Theorem brco3f1o 44023

Description: Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

Hypotheses

Ref	Expression
brco3f1o.c	⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍)
brco3f1o.d	⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌)
brco3f1o.e	⊢ (𝜑 → 𝐸:𝑊–1-1-onto→𝑋)
brco3f1o.r	⊢ (𝜑 → 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)

Assertion

Ref	Expression
brco3f1o	⊢ (𝜑 → ((◡𝐶‘𝐵)𝐶𝐵 ∧ (◡𝐷‘(◡𝐶‘𝐵))𝐷(◡𝐶‘𝐵) ∧ 𝐴𝐸(◡𝐷‘(◡𝐶‘𝐵))))

Proof of Theorem brco3f1o

Step	Hyp	Ref	Expression
1		brco3f1o.e	. . . 4 ⊢ (𝜑 → 𝐸:𝑊–1-1-onto→𝑋)
2		f1ocnv 6861	. . . 4 ⊢ (𝐸:𝑊–1-1-onto→𝑋 → ◡𝐸:𝑋–1-1-onto→𝑊)
3		f1ofn 6850	. . . 4 ⊢ (◡𝐸:𝑋–1-1-onto→𝑊 → ◡𝐸 Fn 𝑋)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → ◡𝐸 Fn 𝑋)
5		brco3f1o.d	. . . 4 ⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌)
6		f1ocnv 6861	. . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → ◡𝐷:𝑌–1-1-onto→𝑋)
7		f1of 6849	. . . 4 ⊢ (◡𝐷:𝑌–1-1-onto→𝑋 → ◡𝐷:𝑌⟶𝑋)
8	5, 6, 7	3syl 18	. . 3 ⊢ (𝜑 → ◡𝐷:𝑌⟶𝑋)
9		brco3f1o.c	. . . 4 ⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍)
10		f1ocnv 6861	. . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → ◡𝐶:𝑍–1-1-onto→𝑌)
11		f1of 6849	. . . 4 ⊢ (◡𝐶:𝑍–1-1-onto→𝑌 → ◡𝐶:𝑍⟶𝑌)
12	9, 10, 11	3syl 18	. . 3 ⊢ (𝜑 → ◡𝐶:𝑍⟶𝑌)
13		brco3f1o.r	. . . 4 ⊢ (𝜑 → 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)
14		relco 6129	. . . . . 6 ⊢ Rel ((𝐶 ∘ 𝐷) ∘ 𝐸)
15	14	relbrcnv 6128	. . . . 5 ⊢ (𝐵◡((𝐶 ∘ 𝐷) ∘ 𝐸)𝐴 ↔ 𝐴((𝐶 ∘ 𝐷) ∘ 𝐸)𝐵)
16		cnvco 5899	. . . . . . 7 ⊢ ◡((𝐶 ∘ 𝐷) ∘ 𝐸) = (◡𝐸 ∘ ◡(𝐶 ∘ 𝐷))
17		cnvco 5899	. . . . . . . 8 ⊢ ◡(𝐶 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐶)
18	17	coeq2i 5874	. . . . . . 7 ⊢ (◡𝐸 ∘ ◡(𝐶 ∘ 𝐷)) = (◡𝐸 ∘ (◡𝐷 ∘ ◡𝐶))
19	16, 18	eqtri 2763	. . . . . 6 ⊢ ◡((𝐶 ∘ 𝐷) ∘ 𝐸) = (◡𝐸 ∘ (◡𝐷 ∘ ◡𝐶))
20	19	breqi 5154	. . . . 5 ⊢ (𝐵◡((𝐶 ∘ 𝐷) ∘ 𝐸)𝐴 ↔ 𝐵(◡𝐸 ∘ (◡𝐷 ∘ ◡𝐶))𝐴)
21		coass 6287	. . . . . 6 ⊢ ((𝐶 ∘ 𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷 ∘ 𝐸))
22	21	breqi 5154	. . . . 5 ⊢ (𝐴((𝐶 ∘ 𝐷) ∘ 𝐸)𝐵 ↔ 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)
23	15, 20, 22	3bitr3ri 302	. . . 4 ⊢ (𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵 ↔ 𝐵(◡𝐸 ∘ (◡𝐷 ∘ ◡𝐶))𝐴)
24	13, 23	sylib 218	. . 3 ⊢ (𝜑 → 𝐵(◡𝐸 ∘ (◡𝐷 ∘ ◡𝐶))𝐴)
25	4, 8, 12, 24	brcofffn 44021	. 2 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷(◡𝐷‘(◡𝐶‘𝐵)) ∧ (◡𝐷‘(◡𝐶‘𝐵))◡𝐸𝐴))
26		f1orel 6852	. . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → Rel 𝐶)
27		relbrcnvg 6126	. . . 4 ⊢ (Rel 𝐶 → (𝐵◡𝐶(◡𝐶‘𝐵) ↔ (◡𝐶‘𝐵)𝐶𝐵))
28	9, 26, 27	3syl 18	. . 3 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ↔ (◡𝐶‘𝐵)𝐶𝐵))
29		f1orel 6852	. . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → Rel 𝐷)
30		relbrcnvg 6126	. . . 4 ⊢ (Rel 𝐷 → ((◡𝐶‘𝐵)◡𝐷(◡𝐷‘(◡𝐶‘𝐵)) ↔ (◡𝐷‘(◡𝐶‘𝐵))𝐷(◡𝐶‘𝐵)))
31	5, 29, 30	3syl 18	. . 3 ⊢ (𝜑 → ((◡𝐶‘𝐵)◡𝐷(◡𝐷‘(◡𝐶‘𝐵)) ↔ (◡𝐷‘(◡𝐶‘𝐵))𝐷(◡𝐶‘𝐵)))
32		f1orel 6852	. . . 4 ⊢ (𝐸:𝑊–1-1-onto→𝑋 → Rel 𝐸)
33		relbrcnvg 6126	. . . 4 ⊢ (Rel 𝐸 → ((◡𝐷‘(◡𝐶‘𝐵))◡𝐸𝐴 ↔ 𝐴𝐸(◡𝐷‘(◡𝐶‘𝐵))))
34	1, 32, 33	3syl 18	. . 3 ⊢ (𝜑 → ((◡𝐷‘(◡𝐶‘𝐵))◡𝐸𝐴 ↔ 𝐴𝐸(◡𝐷‘(◡𝐶‘𝐵))))
35	28, 31, 34	3anbi123d 1435	. 2 ⊢ (𝜑 → ((𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷(◡𝐷‘(◡𝐶‘𝐵)) ∧ (◡𝐷‘(◡𝐶‘𝐵))◡𝐸𝐴) ↔ ((◡𝐶‘𝐵)𝐶𝐵 ∧ (◡𝐷‘(◡𝐶‘𝐵))𝐷(◡𝐶‘𝐵) ∧ 𝐴𝐸(◡𝐷‘(◡𝐶‘𝐵)))))
36	25, 35	mpbid 232	1 ⊢ (𝜑 → ((◡𝐶‘𝐵)𝐶𝐵 ∧ (◡𝐷‘(◡𝐶‘𝐵))𝐷(◡𝐶‘𝐵) ∧ 𝐴𝐸(◡𝐷‘(◡𝐶‘𝐵))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   class class class wbr 5148  ^◡ccnv 5688   ∘ ccom 5693  Rel wrel 5694   Fn wfn 6558  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571

This theorem is referenced by: (None)