Theorem brcoffn 44069

Description: Conditions allowing the decomposition of a binary relation. (Contributed by RP, 7-Jun-2021.)

Hypotheses

Ref	Expression
brcoffn.c	⊢ (𝜑 → 𝐶 Fn 𝑌)
brcoffn.d	⊢ (𝜑 → 𝐷:𝑋⟶𝑌)
brcoffn.r	⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵)

Assertion

Ref	Expression
brcoffn	⊢ (𝜑 → (𝐴𝐷(𝐷‘𝐴) ∧ (𝐷‘𝐴)𝐶𝐵))

Proof of Theorem brcoffn

Step	Hyp	Ref	Expression
1		brcoffn.c	. . . 4 ⊢ (𝜑 → 𝐶 Fn 𝑌)
2		brcoffn.d	. . . 4 ⊢ (𝜑 → 𝐷:𝑋⟶𝑌)
3		fnfco 6688	. . . 4 ⊢ ((𝐶 Fn 𝑌 ∧ 𝐷:𝑋⟶𝑌) → (𝐶 ∘ 𝐷) Fn 𝑋)
4	1, 2, 3	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐶 ∘ 𝐷) Fn 𝑋)
5		simpl 482	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋) → 𝜑)
6		simpr 484	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋) → (𝐶 ∘ 𝐷) Fn 𝑋)
7		brcoffn.r	. . . . . 6 ⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵)
8	5, 7	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋) → 𝐴(𝐶 ∘ 𝐷)𝐵)
9		fnbr 6589	. . . . 5 ⊢ (((𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴(𝐶 ∘ 𝐷)𝐵) → 𝐴 ∈ 𝑋)
10	6, 8, 9	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋) → 𝐴 ∈ 𝑋)
11	5, 6, 10	3jca 1128	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋) → (𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋))
12	4, 11	mpdan 687	. 2 ⊢ (𝜑 → (𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋))
13	2	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐷:𝑋⟶𝑌)
14		simp3 1138	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
15		fvco3 6921	. . . . . 6 ⊢ ((𝐷:𝑋⟶𝑌 ∧ 𝐴 ∈ 𝑋) → ((𝐶 ∘ 𝐷)‘𝐴) = (𝐶‘(𝐷‘𝐴)))
16	13, 14, 15	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐶 ∘ 𝐷)‘𝐴) = (𝐶‘(𝐷‘𝐴)))
17	7	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴(𝐶 ∘ 𝐷)𝐵)
18		fnbrfvb 6872	. . . . . . 7 ⊢ (((𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝐶 ∘ 𝐷)‘𝐴) = 𝐵 ↔ 𝐴(𝐶 ∘ 𝐷)𝐵))
19	18	3adant1 1130	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝐶 ∘ 𝐷)‘𝐴) = 𝐵 ↔ 𝐴(𝐶 ∘ 𝐷)𝐵))
20	17, 19	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐶 ∘ 𝐷)‘𝐴) = 𝐵)
21	16, 20	eqtr3d 2768	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐶‘(𝐷‘𝐴)) = 𝐵)
22		eqid 2731	. . . 4 ⊢ (𝐷‘𝐴) = (𝐷‘𝐴)
23	21, 22	jctil 519	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐷‘𝐴) = (𝐷‘𝐴) ∧ (𝐶‘(𝐷‘𝐴)) = 𝐵))
24	13	ffnd 6652	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐷 Fn 𝑋)
25		fnbrfvb 6872	. . . . 5 ⊢ ((𝐷 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐷‘𝐴) = (𝐷‘𝐴) ↔ 𝐴𝐷(𝐷‘𝐴)))
26	24, 14, 25	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐷‘𝐴) = (𝐷‘𝐴) ↔ 𝐴𝐷(𝐷‘𝐴)))
27	1	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐶 Fn 𝑌)
28	13, 14	ffvelcdmd 7018	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐷‘𝐴) ∈ 𝑌)
29		fnbrfvb 6872	. . . . 5 ⊢ ((𝐶 Fn 𝑌 ∧ (𝐷‘𝐴) ∈ 𝑌) → ((𝐶‘(𝐷‘𝐴)) = 𝐵 ↔ (𝐷‘𝐴)𝐶𝐵))
30	27, 28, 29	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐶‘(𝐷‘𝐴)) = 𝐵 ↔ (𝐷‘𝐴)𝐶𝐵))
31	26, 30	anbi12d 632	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝐷‘𝐴) = (𝐷‘𝐴) ∧ (𝐶‘(𝐷‘𝐴)) = 𝐵) ↔ (𝐴𝐷(𝐷‘𝐴) ∧ (𝐷‘𝐴)𝐶𝐵)))
32	23, 31	mpbid 232	. 2 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷(𝐷‘𝐴) ∧ (𝐷‘𝐴)𝐶𝐵))
33	12, 32	syl 17	1 ⊢ (𝜑 → (𝐴𝐷(𝐷‘𝐴) ∧ (𝐷‘𝐴)𝐶𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   class class class wbr 5091   ∘ ccom 5620   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489

This theorem is referenced by:  brcofffn  44070  brco2f1o  44071  clsneikex  44145  clsneinex  44146  clsneiel1  44147