Theorem brcoffn 44020

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 6774	. . . 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 6677	. . . . 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 687	. 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 7008	. . . . . 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 6960	. . . . . . 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 257	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐶 ∘ 𝐷)‘𝐴) = 𝐵)
21	16, 20	eqtr3d 2777	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐶‘(𝐷‘𝐴)) = 𝐵)
22		eqid 2735	. . . 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 6738	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐷 Fn 𝑋)
25		fnbrfvb 6960	. . . . 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	ffvelcdmd 7105	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∘ 𝐷) Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐷‘𝐴) ∈ 𝑌)
29		fnbrfvb 6960	. . . . 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 1537   ∈ wcel 2106   class class class wbr 5148   ∘ ccom 5693   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571

This theorem is referenced by:  brcofffn  44021  brco2f1o  44022  clsneikex  44096  clsneinex  44097  clsneiel1  44098