Theorem brcofffn 44214

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

Hypotheses

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

Assertion

Ref	Expression
brcofffn	⊢ (𝜑 → (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵))

Proof of Theorem brcofffn

Step	Hyp	Ref	Expression
1		brcofffn.c	. . . . 5 ⊢ (𝜑 → 𝐶 Fn 𝑍)
2		brcofffn.d	. . . . 5 ⊢ (𝜑 → 𝐷:𝑌⟶𝑍)
3		fnfco 6697	. . . . 5 ⊢ ((𝐶 Fn 𝑍 ∧ 𝐷:𝑌⟶𝑍) → (𝐶 ∘ 𝐷) Fn 𝑌)
4	1, 2, 3	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐶 ∘ 𝐷) Fn 𝑌)
5		brcofffn.e	. . . 4 ⊢ (𝜑 → 𝐸:𝑋⟶𝑌)
6		brcofffn.r	. . . . 5 ⊢ (𝜑 → 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)
7		coass 6222	. . . . . 6 ⊢ ((𝐶 ∘ 𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷 ∘ 𝐸))
8	7	breqi 5102	. . . . 5 ⊢ (𝐴((𝐶 ∘ 𝐷) ∘ 𝐸)𝐵 ↔ 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)
9	6, 8	sylibr 234	. . . 4 ⊢ (𝜑 → 𝐴((𝐶 ∘ 𝐷) ∘ 𝐸)𝐵)
10	4, 5, 9	brcoffn 44213	. . 3 ⊢ (𝜑 → (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵))
11	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵)) → 𝐶 Fn 𝑍)
12	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵)) → 𝐷:𝑌⟶𝑍)
13		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵)) → (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵)
14	11, 12, 13	brcoffn 44213	. . . 4 ⊢ ((𝜑 ∧ (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵)) → ((𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵))
15	14	ex 412	. . 3 ⊢ (𝜑 → ((𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵) → ((𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵)))
16	10, 15	jcai 516	. 2 ⊢ (𝜑 → ((𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵) ∧ ((𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵)))
17		simpll 766	. . 3 ⊢ (((𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵) ∧ ((𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵)) → 𝐴𝐸(𝐸‘𝐴))
18		simprl 770	. . 3 ⊢ (((𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵) ∧ ((𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵)) → (𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)))
19		simprr 772	. . 3 ⊢ (((𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵) ∧ ((𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵)) → (𝐷‘(𝐸‘𝐴))𝐶𝐵)
20	17, 18, 19	3jca 1128	. 2 ⊢ (((𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)(𝐶 ∘ 𝐷)𝐵) ∧ ((𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵)) → (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵))
21	16, 20	syl 17	1 ⊢ (𝜑 → (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   class class class wbr 5096   ∘ ccom 5626   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498

This theorem is referenced by:  brco3f1o  44216  neicvgmex  44300  neicvgel1  44302