Theorem brcoffn 41640

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 6639	. . . 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 483	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋) → 𝜑)
6		simpr 485	. . . 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 6541	. . . . 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 1127	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋) → (𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋))
12	4, 11	mpdan 684	. 2 ⊢ (𝜑 → (𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋))
13	2	3ad2ant1 1132	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐷:𝑋⟶𝑌)
14		simp3 1137	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
15		fvco3 6867	. . . . . 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 1132	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴(𝐶 ∘ 𝐷)𝐵)
18		fnbrfvb 6822	. . . . . . 7 ⊢ (((𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝐶 ∘ 𝐷)‘𝐴) = 𝐵 ↔ 𝐴(𝐶 ∘ 𝐷)𝐵))
19	18	3adant1 1129	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝐶 ∘ 𝐷)‘𝐴) = 𝐵 ↔ 𝐴(𝐶 ∘ 𝐷)𝐵))
20	17, 19	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐶 ∘ 𝐷)‘𝐴) = 𝐵)
21	16, 20	eqtr3d 2780	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐶‘(𝐷‘𝐴)) = 𝐵)
22		eqid 2738	. . . 4 ⊢ (𝐷‘𝐴) = (𝐷‘𝐴)
23	21, 22	jctil 520	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐷‘𝐴) = (𝐷‘𝐴) ∧ (𝐶‘(𝐷‘𝐴)) = 𝐵))
24	13	ffnd 6601	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐷 Fn 𝑋)
25		fnbrfvb 6822	. . . . 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 1132	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐶 Fn 𝑌)
28	13, 14	ffvelrnd 6962	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐷‘𝐴) ∈ 𝑌)
29		fnbrfvb 6822	. . . . 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 631	. . 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 231	. 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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   class class class wbr 5074   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  brcofffn  41641  brco2f1o  41642  clsneikex  41716  clsneinex  41717  clsneiel1  41718