Theorem rescofuf 50533

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 3458	. . . . 5 ⊢ 𝑔 ∈ V
2		vex 3458	. . . . 5 ⊢ 𝑓 ∈ V
3		opex 5437	. . . . 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 18076	. . . . . 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 7597	. . . . 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 1491	. . . 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 18104	. . . 4 ⊢ ((𝑔 ∈ (𝐷 Func 𝐸) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (𝑔 ∘func 𝑓) ∈ (𝐶 Func 𝐸))
10	6, 9	eqeltrrid 2869	. . 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 3206	. 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 5980	. . . 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 3937	. . . . 5 ⊢ (𝐷 Func 𝐸) ⊆ V
14		ssv 3937	. . . . 5 ⊢ (𝐶 Func 𝐷) ⊆ V
15		resmpo 7569	. . . . 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 2786	. . 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 8112	. 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 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ⟨cop 4838   × cxp 5654  dom cdm 5656   ↾ cres 5658   ∘ ccom 5660  ⟶wf 6560  ‘cfv 6564  (class class class)co 7449   ∈ cmpo 7451  1^st c1st 8031  2^nd c2nd 8032   Func cfunc 18070   ∘_func ccofu 18072

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-1st 8033  df-2nd 8034  df-map 8885  df-ixp 8962  df-cat 17883  df-cid 17884  df-func 18074  df-cofu 18076

This theorem is used by:  fucof1  50769  fucofvalne  50772  fucoisfunc  50812  fucoisfunc2lem1  50813