Theorem ofrco 32562

Description: Function relation between function compositions. (Contributed by Thierry Arnoux, 15-Jan-2026.)

Hypotheses

Ref	Expression
ofrco.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
ofrco.2	⊢ (𝜑 → 𝐺 Fn 𝐴)
ofrco.3	⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
ofrco.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofrco.5	⊢ (𝜑 → 𝐶 ∈ 𝑊)
ofrco.6	⊢ (𝜑 → 𝐹 ∘r 𝑅𝐺)

Assertion

Ref	Expression
ofrco	⊢ (𝜑 → (𝐹 ∘ 𝐻) ∘r 𝑅(𝐺 ∘ 𝐻))

Proof of Theorem ofrco

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6822	. . . . 5 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
2		fveq2 6822	. . . . 5 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
3	1, 2	breq12d 5105	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → ((𝐹‘𝑦)𝑅(𝐺‘𝑦) ↔ (𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
4		ofrco.6	. . . . . 6 ⊢ (𝜑 → 𝐹 ∘r 𝑅𝐺)
5		ofrco.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
6		ofrco.2	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐴)
7		ofrco.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		inidm 4178	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9		eqidd 2730	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
10		eqidd 2730	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
11	5, 6, 7, 7, 8, 9, 10	ofrfval 7623	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦)𝑅(𝐺‘𝑦)))
12	4, 11	mpbid 232	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦)𝑅(𝐺‘𝑦))
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦)𝑅(𝐺‘𝑦))
14		ofrco.3	. . . . 5 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
15	14	ffvelcdmda 7018	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝐴)
16	3, 13, 15	rspcdva 3578	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))
17	16	ralrimiva 3121	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 (𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))
18		fnfco 6689	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐻:𝐶⟶𝐴) → (𝐹 ∘ 𝐻) Fn 𝐶)
19	5, 14, 18	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐻) Fn 𝐶)
20		fnfco 6689	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐻:𝐶⟶𝐴) → (𝐺 ∘ 𝐻) Fn 𝐶)
21	6, 14, 20	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐻) Fn 𝐶)
22		ofrco.5	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
23		inidm 4178	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
24	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐻:𝐶⟶𝐴)
25		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
26	24, 25	fvco3d 6923	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
27	24, 25	fvco3d 6923	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
28	19, 21, 22, 22, 23, 26, 27	ofrfval 7623	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ∘r 𝑅(𝐺 ∘ 𝐻) ↔ ∀𝑥 ∈ 𝐶 (𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
29	17, 28	mpbird 257	1 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∘r 𝑅(𝐺 ∘ 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5092   ∘ ccom 5623   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482   ∘_r cofr 7612

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-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

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-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ofr 7614

This theorem is referenced by:  mplvrpmrhm  33558