Theorem ofrco 33070

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 6882	. . . . 5 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
2		fveq2 6882	. . . . 5 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
3	1, 2	breq12d 5120	. . . 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 4175	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9		eqidd 2763	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
10		eqidd 2763	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
11	5, 6, 7, 7, 8, 9, 10	ofrfval 7691	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦)𝑅(𝐺‘𝑦)))
12	4, 11	mpbid 235	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦)𝑅(𝐺‘𝑦))
13	12	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦)𝑅(𝐺‘𝑦))
14		ofrco.3	. . . . 5 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
15	14	ffvelcdmda 7080	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝐴)
16	3, 13, 15	rspcdva 3580	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))
17	16	ralrimiva 3156	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 (𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))
18		fnfco 6744	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐻:𝐶⟶𝐴) → (𝐹 ∘ 𝐻) Fn 𝐶)
19	5, 14, 18	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐻) Fn 𝐶)
20		fnfco 6744	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐻:𝐶⟶𝐴) → (𝐺 ∘ 𝐻) Fn 𝐶)
21	6, 14, 20	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐻) Fn 𝐶)
22		ofrco.5	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
23		inidm 4175	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
24	14	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐻:𝐶⟶𝐴)
25		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
26	24, 25	fvco3d 6983	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
27	24, 25	fvco3d 6983	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
28	19, 21, 22, 22, 23, 26, 27	ofrfval 7691	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ∘r 𝑅(𝐺 ∘ 𝐻) ↔ ∀𝑥 ∈ 𝐶 (𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
29	17, 28	mpbird 260	1 ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∘r 𝑅(𝐺 ∘ 𝐻))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078   class class class wbr 5107   ∘ ccom 5663   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537   ∘_r cofr 7680

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ofr 7682

This theorem is used by:  mplvrpmrhm  34044