Theorem prcof21a 49164

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 17953	. . . . 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 49162	. 2 ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))
10		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
11	10	coeq1d 5839	. 2 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑎 ∘ (1st ‘𝐹)) = (𝐴 ∘ (1st ‘𝐹)))
12		fvexd 6888	. . 3 ⊢ (𝜑 → (1st ‘𝐹) ∈ V)
13	2, 12	coexd 7922	. 2 ⊢ (𝜑 → (𝐴 ∘ (1st ‘𝐹)) ∈ V)
14	9, 11, 2, 13	fvmptd 6990	1 ⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3457  ⟨cop 4605   ∘ ccom 5656  ‘cfv 6528  (class class class)co 7400  1^st c1st 7981  2^nd c2nd 7982   Func cfunc 17854   Nat cnat 17944   −∘_F cprcof 49147

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 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

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-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7983  df-2nd 7984  df-ixp 8907  df-func 17858  df-nat 17946  df-prcof 49148

This theorem is referenced by:  prcof22a  49165  lanup  49376  ranup  49377