Theorem rescofuf 48869

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 3467	. . . . 5 ⊢ 𝑔 ∈ V
2		vex 3467	. . . . 5 ⊢ 𝑓 ∈ V
3		opex 5449	. . . . 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 17875	. . . . . 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 7561	. . . . 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 1462	. . . 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 484	. . . . 5 ⊢ ((𝑔 ∈ (𝐷 Func 𝐸) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑓 ∈ (𝐶 Func 𝐷))
8		simpl 482	. . . . 5 ⊢ ((𝑔 ∈ (𝐷 Func 𝐸) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → 𝑔 ∈ (𝐷 Func 𝐸))
9	7, 8	cofucl 17903	. . . 4 ⊢ ((𝑔 ∈ (𝐷 Func 𝐸) ∧ 𝑓 ∈ (𝐶 Func 𝐷)) → (𝑔 ∘func 𝑓) ∈ (𝐶 Func 𝐸))
10	6, 9	eqeltrrid 2838	. . 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 3186	. 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 5973	. . . 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 3988	. . . . 5 ⊢ (𝐷 Func 𝐸) ⊆ V
14		ssv 3988	. . . . 5 ⊢ (𝐶 Func 𝐷) ⊆ V
15		resmpo 7534	. . . . 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 692	. . . 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 2757	. . 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 8074	. 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 230	1 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸)

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  Vcvv 3463   ⊆ wss 3931  ⟨cop 4612   × cxp 5663  dom cdm 5665   ↾ cres 5667   ∘ ccom 5669  ⟶wf 6536  ‘cfv 6540  (class class class)co 7412   ∈ cmpo 7414  1^st c1st 7993  2^nd c2nd 7994   Func cfunc 17869   ∘_func ccofu 17871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7995  df-2nd 7996  df-map 8849  df-ixp 8919  df-cat 17681  df-cid 17682  df-func 17873  df-cofu 17875

This theorem is referenced by:  fucof1  48969  fucofvalne  48972