Theorem brco2f1o 41881

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 6765	. . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → ◡𝐷:𝑌–1-1-onto→𝑋)
3		f1ofn 6754	. . . 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 6765	. . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → ◡𝐶:𝑍–1-1-onto→𝑌)
7		f1of 6753	. . . 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 6033	. . . . . 6 ⊢ Rel (𝐶 ∘ 𝐷)
11	10	relbrcnv 6032	. . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐴(𝐶 ∘ 𝐷)𝐵)
12		cnvco 5814	. . . . . 6 ⊢ ◡(𝐶 ∘ 𝐷) = (◡𝐷 ∘ ◡𝐶)
13	12	breqi 5092	. . . . 5 ⊢ (𝐵◡(𝐶 ∘ 𝐷)𝐴 ↔ 𝐵(◡𝐷 ∘ ◡𝐶)𝐴)
14	11, 13	bitr3i 276	. . . 4 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵(◡𝐷 ∘ ◡𝐶)𝐴)
15	9, 14	sylib 217	. . 3 ⊢ (𝜑 → 𝐵(◡𝐷 ∘ ◡𝐶)𝐴)
16	4, 8, 15	brcoffn 41879	. 2 ⊢ (𝜑 → (𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷𝐴))
17		f1orel 6756	. . . 4 ⊢ (𝐶:𝑌–1-1-onto→𝑍 → Rel 𝐶)
18		relbrcnvg 6030	. . . 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 6756	. . . 4 ⊢ (𝐷:𝑋–1-1-onto→𝑌 → Rel 𝐷)
21		relbrcnvg 6030	. . . 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 631	. 2 ⊢ (𝜑 → ((𝐵◡𝐶(◡𝐶‘𝐵) ∧ (◡𝐶‘𝐵)◡𝐷𝐴) ↔ ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵))))
24	16, 23	mpbid 231	1 ⊢ (𝜑 → ((◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(◡𝐶‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   class class class wbr 5086  ^◡ccnv 5606   ∘ ccom 5611  Rel wrel 5612   Fn wfn 6460  ⟶wf 6461  –1-1-onto→wf1o 6464  ‘cfv 6465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pr 5366

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-br 5087  df-opab 5149  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473

This theorem is referenced by: (None)