Theorem ofrco 32593

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 5102	. . . 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 4174	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9		eqidd 2732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
10		eqidd 2732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
11	5, 6, 7, 7, 8, 9, 10	ofrfval 7620	. . . . . 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 7017	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝐴)
16	3, 13, 15	rspcdva 3573	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))
17	16	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 (𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))
18		fnfco 6688	. . . 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 6688	. . . 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 4174	. . 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 6922	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
27	24, 25	fvco3d 6922	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
28	19, 21, 22, 22, 23, 26, 27	ofrfval 7620	. 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 1541   ∈ wcel 2111  ∀wral 3047   class class class wbr 5089   ∘ ccom 5618   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481   ∘_r cofr 7609

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ofr 7611

This theorem is referenced by:  mplvrpmrhm  33577