Theorem prcof21a 49373

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 17895	. . . . 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 49371	. 2 ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))
10		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
11	10	coeq1d 5815	. 2 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑎 ∘ (1st ‘𝐹)) = (𝐴 ∘ (1st ‘𝐹)))
12		fvexd 6855	. . 3 ⊢ (𝜑 → (1st ‘𝐹) ∈ V)
13	2, 12	coexd 7887	. 2 ⊢ (𝜑 → (𝐴 ∘ (1st ‘𝐹)) ∈ V)
14	9, 11, 2, 13	fvmptd 6957	1 ⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591   ∘ ccom 5635  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946   Func cfunc 17796   Nat cnat 17886   −∘_F cprcof 49355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-ixp 8848  df-func 17800  df-nat 17888  df-prcof 49356

This theorem is referenced by:  prcof22a  49374  prcofdiag  49376  lanup  49623  ranup  49624