Theorem prcof21a 49502

Description: The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)

Hypotheses

Ref	Expression
prcof21a.n	⊢ 𝑁 = (𝐷 Nat 𝐸)
prcof21a.a	⊢ (𝜑 → 𝐴 ∈ (𝐾𝑁𝐿))
prcof21a.p	⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)
prcof21a.f	⊢ (𝜑 → 𝐹 ∈ 𝑈)

Assertion

Ref	Expression
prcof21a	⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))

Proof of Theorem prcof21a

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prcof21a.n	. . 3 ⊢ 𝑁 = (𝐷 Nat 𝐸)
2		prcof21a.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐾𝑁𝐿))
3	1	natrcl 17860	. . . . 5 ⊢ (𝐴 ∈ (𝐾𝑁𝐿) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝐿 ∈ (𝐷 Func 𝐸)))
4	2, 3	syl 17	. . . 4 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝐿 ∈ (𝐷 Func 𝐸)))
5	4	simpld 494	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
6	4	simprd 495	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
7		prcof21a.p	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)
8		prcof21a.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑈)
9	1, 5, 6, 7, 8	prcof2a 49500	. 2 ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))
10		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
11	10	coeq1d 5800	. 2 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑎 ∘ (1st ‘𝐹)) = (𝐴 ∘ (1st ‘𝐹)))
12		fvexd 6837	. . 3 ⊢ (𝜑 → (1st ‘𝐹) ∈ V)
13	2, 12	coexd 7861	. 2 ⊢ (𝜑 → (𝐴 ∘ (1st ‘𝐹)) ∈ V)
14	9, 11, 2, 13	fvmptd 6936	1 ⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4579   ∘ ccom 5618  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920   Func cfunc 17761   Nat cnat 17851   −∘_F cprcof 49484

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-pow 5301  ax-pr 5368  ax-un 7668

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-pw 4549  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-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-ixp 8822  df-func 17765  df-nat 17853  df-prcof 49485

This theorem is referenced by:  prcof22a  49503  prcofdiag  49505  lanup  49752  ranup  49753