Theorem rescofuf 50019

Description: The restriction of functor composition is a function from product functor space to functor space. (Contributed by Zhi Wang, 25-Sep-2025.)

Assertion

Ref	Expression
rescofuf	⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸)

Proof of Theorem rescofuf

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

Step	Hyp	Ref	Expression
1		vex 3454	. . . . 5 ⊢ 𝑔 ∈ V
2		vex 3454	. . . . 5 ⊢ 𝑓 ∈ V
3		opex 5439	. . . . 5 ⊢ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩ ∈ V
4		df-cofu 17949	. . . . . 6 ⊢ ∘func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩)
5	4	ovmpt4g 7560	. . . . 5 ⊢ ((𝑔 ∈ V ∧ 𝑓 ∈ V ∧ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩ ∈ V) → (𝑔 ∘func 𝑓) = ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩)
6	1, 2, 3, 5	mp3an 1490	. . . 4 ⊢ (𝑔 ∘func 𝑓) = ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩
7		simpr 490	. . . . 5 ⊢ ((𝑔 ∈ (𝐷 Func 𝐸) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑓 ∈ (𝐶 Func 𝐷))
8		simpl 488	. . . . 5 ⊢ ((𝑔 ∈ (𝐷 Func 𝐸) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑔 ∈ (𝐷 Func 𝐸))
9	7, 8	cofucl 17977	. . . 4 ⊢ ((𝑔 ∈ (𝐷 Func 𝐸) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (𝑔 ∘func 𝑓) ∈ (𝐶 Func 𝐸))
10	6, 9	eqeltrrid 2865	. . 3 ⊢ ((𝑔 ∈ (𝐷 Func 𝐸) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩ ∈ (𝐶 Func 𝐸))
11	10	rgen2 3202	. 2 ⊢ ∀𝑔 ∈ (𝐷 Func 𝐸)∀𝑓 ∈ (𝐶 Func 𝐷)⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩ ∈ (𝐶 Func 𝐸)
12	4	reseq1i 5968	. . . 4 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) = ((𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩) ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
13		ssv 3955	. . . . 5 ⊢ (𝐷 Func 𝐸) ⊆ V
14		ssv 3955	. . . . 5 ⊢ (𝐶 Func 𝐷) ⊆ V
15		resmpo 7533	. . . . 5 ⊢ (((𝐷 Func 𝐸) ⊆ V ∧ (𝐶 Func 𝐷) ⊆ V) → ((𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩) ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) = (𝑔 ∈ (𝐷 Func 𝐸), 𝑓 ∈ (𝐶 Func 𝐷) ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩))
16	13, 14, 15	mp2an 705	. . . 4 ⊢ ((𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩) ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) = (𝑔 ∈ (𝐷 Func 𝐸), 𝑓 ∈ (𝐶 Func 𝐷) ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩)
17	12, 16	eqtri 2783	. . 3 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) = (𝑔 ∈ (𝐷 Func 𝐸), 𝑓 ∈ (𝐶 Func 𝐷) ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩)
18	17	fmpo 8065	. 2 ⊢ (∀𝑔 ∈ (𝐷 Func 𝐸)∀𝑓 ∈ (𝐶 Func 𝐷)⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩ ∈ (𝐶 Func 𝐸) ↔ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸))
19	11, 18	mpbi 233	1 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3899  ⟨cop 4590   × cxp 5653  dom cdm 5655   ↾ cres 5657   ∘ ccom 5659  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7984  2^nd c2nd 7985   Func cfunc 17943   ∘_func ccofu 17945

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 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-map 8828  df-ixp 8905  df-cat 17756  df-cid 17757  df-func 17947  df-cofu 17949

This theorem is used by:  fucof1  50248  fucofvalne  50251