Theorem brco2f1o 43994

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

Hypotheses

Ref	Expression
brco2f1o.c	⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍)
brco2f1o.d	⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌)
brco2f1o.r	⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵)

Assertion

Ref	Expression
brco2f1o	⊢ (𝜑 → ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵)))

Proof of Theorem brco2f1o

Step	Hyp	Ref	Expression
1		brco2f1o.d	. . . 4 ⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌)
2		f1ocnv 6794	. . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → ◡𝐷:𝑌–1-1-onto→𝑋)
3		f1ofn 6783	. . . 4 ⊢ (◡𝐷:𝑌–1-1-onto→𝑋 → ◡𝐷 Fn 𝑌)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → ◡𝐷 Fn 𝑌)
5		brco2f1o.c	. . . 4 ⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍)
6		f1ocnv 6794	. . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → ◡𝐶:𝑍–1-1-onto→𝑌)
7		f1of 6782	. . . 4 ⊢ (◡𝐶:𝑍–1-1-onto→𝑌 → ◡𝐶:𝑍⟶𝑌)
8	5, 6, 7	3syl 18	. . 3 ⊢ (𝜑 → ◡𝐶:𝑍⟶𝑌)
9		brco2f1o.r	. . . 4 ⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵)
10		relco 6068	. . . . . 6 ⊢ Rel (𝐶 ∘ 𝐷)
11	10	relbrcnv 6067	. . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐴(𝐶 ∘ 𝐷)𝐵)
12		cnvco 5839	. . . . . 6 ⊢ ◡(𝐶 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐶)
13	12	breqi 5108	. . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐵(◡𝐷 ∘ ◡𝐶)𝐴)
14	11, 13	bitr3i 277	. . . 4 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵(◡𝐷 ∘ ◡𝐶)𝐴)
15	9, 14	sylib 218	. . 3 ⊢ (𝜑 → 𝐵(◡𝐷 ∘ ◡𝐶)𝐴)
16	4, 8, 15	brcoffn 43992	. 2 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷𝐴))
17		f1orel 6785	. . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → Rel 𝐶)
18		relbrcnvg 6065	. . . 4 ⊢ (Rel 𝐶 → (𝐵◡𝐶(◡𝐶‘𝐵) ↔ (◡𝐶‘𝐵)𝐶𝐵))
19	5, 17, 18	3syl 18	. . 3 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ↔ (◡𝐶‘𝐵)𝐶𝐵))
20		f1orel 6785	. . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → Rel 𝐷)
21		relbrcnvg 6065	. . . 4 ⊢ (Rel 𝐷 → ((◡𝐶‘𝐵)◡𝐷𝐴 ↔ 𝐴𝐷(◡𝐶‘𝐵)))
22	1, 20, 21	3syl 18	. . 3 ⊢ (𝜑 → ((◡𝐶‘𝐵)◡𝐷𝐴 ↔ 𝐴𝐷(◡𝐶‘𝐵)))
23	19, 22	anbi12d 632	. 2 ⊢ (𝜑 → ((𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷𝐴) ↔ ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵))))
24	16, 23	mpbid 232	1 ⊢ (𝜑 → ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   class class class wbr 5102  ^◡ccnv 5630   ∘ ccom 5635  Rel wrel 5636   Fn wfn 6494  ⟶wf 6495  –1-1-onto→wf1o 6498  ‘cfv 6499

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507

This theorem is referenced by: (None)