Theorem brco3f1o 44570

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 6814	. . . 4 ⊢ (𝐸:𝑊–1-1-onto→𝑋 → ◡𝐸:𝑋–1-1-onto→𝑊)
3		f1ofn 6802	. . . 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 6814	. . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → ◡𝐷:𝑌–1-1-onto→𝑋)
7		f1of 6801	. . . 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 6814	. . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → ◡𝐶:𝑍–1-1-onto→𝑌)
11		f1of 6801	. . . 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 6093	. . . . . 6 ⊢ Rel ((𝐶 ∘ 𝐷) ∘ 𝐸)
15	14	relbrcnv 6092	. . . . 5 ⊢ (𝐵◡((𝐶 ∘ 𝐷) ∘ 𝐸)𝐴 ↔ 𝐴((𝐶 ∘ 𝐷) ∘ 𝐸)𝐵)
16		cnvco 5857	. . . . . . 7 ⊢ ◡((𝐶 ∘ 𝐷) ∘ 𝐸) = (◡𝐸 ∘ ◡(𝐶 ∘ 𝐷))
17		cnvco 5857	. . . . . . . 8 ⊢ ◡(𝐶 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐶)
18	17	coeq2i 5828	. . . . . . 7 ⊢ (◡𝐸 ∘ ◡(𝐶 ∘ 𝐷)) = (◡𝐸 ∘ (◡𝐷 ∘ ◡𝐶))
19	16, 18	eqtri 2784	. . . . . 6 ⊢ ◡((𝐶 ∘ 𝐷) ∘ 𝐸) = (◡𝐸 ∘ (◡𝐷 ∘ ◡𝐶))
20	19	breqi 5103	. . . . 5 ⊢ (𝐵◡((𝐶 ∘ 𝐷) ∘ 𝐸)𝐴 ↔ 𝐵(◡𝐸 ∘ (◡𝐷 ∘ ◡𝐶))𝐴)
21		coass 6248	. . . . . 6 ⊢ ((𝐶 ∘ 𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷 ∘ 𝐸))
22	21	breqi 5103	. . . . 5 ⊢ (𝐴((𝐶 ∘ 𝐷) ∘ 𝐸)𝐵 ↔ 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)
23	15, 20, 22	3bitr3ri 304	. . . 4 ⊢ (𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵 ↔ 𝐵(◡𝐸 ∘ (◡𝐷 ∘ ◡𝐶))𝐴)
24	13, 23	sylib 220	. . 3 ⊢ (𝜑 → 𝐵(◡𝐸 ∘ (◡𝐷 ∘ ◡𝐶))𝐴)
25	4, 8, 12, 24	brcofffn 44568	. 2 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷(◡𝐷‘(◡𝐶‘𝐵)) ∧ (◡𝐷‘(◡𝐶‘𝐵))◡𝐸𝐴))
26		f1orel 6804	. . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → Rel 𝐶)
27		relbrcnvg 6090	. . . 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 6804	. . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → Rel 𝐷)
30		relbrcnvg 6090	. . . 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 6804	. . . 4 ⊢ (𝐸:𝑊–1-1-onto→𝑋 → Rel 𝐸)
33		relbrcnvg 6090	. . . 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 1456	. 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 234	1 ⊢ (𝜑 → ((◡𝐶‘𝐵)𝐶𝐵 ∧ (◡𝐷‘(◡𝐶‘𝐵))𝐷(◡𝐶‘𝐵) ∧ 𝐴𝐸(◡𝐷‘(◡𝐶‘𝐵))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1097   class class class wbr 5097  ^◡ccnv 5642   ∘ ccom 5647  Rel wrel 5648   Fn wfn 6511  ⟶wf 6512  –1-1-onto→wf1o 6515  ‘cfv 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524

This theorem is referenced by: (None)