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Theorem prcof22a 50014
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 (𝜑𝐴 ∈ (𝐾𝑁𝐿))
prcof21a.p (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)
prcof22a.b 𝐵 = (Base‘𝐶)
prcof22a.x (𝜑𝑋𝐵)
prcof22a.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
Assertion
Ref Expression
prcof22a (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = (𝐴‘((1st𝐹)‘𝑋)))

Proof of Theorem prcof22a
StepHypRef Expression
1 prcof21a.n . . . 4 𝑁 = (𝐷 Nat 𝐸)
2 prcof21a.a . . . 4 (𝜑𝐴 ∈ (𝐾𝑁𝐿))
3 prcof21a.p . . . 4 (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)
4 prcof22a.f . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
51, 2, 3, 4prcof21a 50013 . . 3 (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st𝐹)))
65fveq1d 6870 . 2 (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = ((𝐴 ∘ (1st𝐹))‘𝑋))
7 prcof22a.b . . . 4 𝐵 = (Base‘𝐶)
8 eqid 2763 . . . 4 (Base‘𝐷) = (Base‘𝐷)
94func1st2nd 49698 . . . 4 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
107, 8, 9funcf1 17900 . . 3 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
11 prcof22a.x . . 3 (𝜑𝑋𝐵)
1210, 11fvco3d 6969 . 2 (𝜑 → ((𝐴 ∘ (1st𝐹))‘𝑋) = (𝐴‘((1st𝐹)‘𝑋)))
136, 12eqtrd 2798 1 (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = (𝐴‘((1st𝐹)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1561  wcel 2143  cop 4589  ccom 5652  cfv 6522  (class class class)co 7397  1st c1st 7969  2nd c2nd 7970  Basecbs 17246   Func cfunc 17888   Nat cnat 17978   −∘F cprcof 49995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-1st 7971  df-2nd 7972  df-map 8811  df-ixp 8881  df-func 17892  df-nat 17980  df-prcof 49996
This theorem is referenced by: (None)
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