Theorem brcofffn 43993

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 6707	. . . . 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 6226	. . . . . 6 ⊢ ((𝐶 ∘ 𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷 ∘ 𝐸))
8	7	breqi 5108	. . . . 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 43992	. . 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 43992	. . . 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 5102   ∘ ccom 5635   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507

This theorem is referenced by:  brco3f1o  43995  neicvgmex  44079  neicvgel1  44081