Theorem cofu1a 49723

Description: Value of the object part of the functor composition. (Contributed by Zhi Wang, 16-Nov-2025.)

Hypotheses

Ref	Expression
cofu1a.b	⊢ 𝐵 = (Base‘𝐶)
cofu1a.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
cofu1a.k	⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
cofu1a.m	⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩)
cofu1a.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
cofu1a	⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = (𝑀‘𝑋))

Proof of Theorem cofu1a

Step	Hyp	Ref	Expression
1		cofu1a.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		cofu1a.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
3		df-br 5106	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4	2, 3	sylib 221	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5		cofu1a.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
6		df-br 5106	. . . 4 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
7	5, 6	sylib 221	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
8		cofu1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	1, 4, 7, 8	cofu1 17931	. 2 ⊢ (𝜑 → ((1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))‘𝑋) = ((1st ‘⟨𝐾, 𝐿⟩)‘((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)))
10		cofu1a.m	. . . . 5 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩)
11	10	fveq2d 6875	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = (1st ‘⟨𝑀, 𝑁⟩))
12	4, 7	cofucl 17935	. . . . . . 7 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
13	10, 12	eqeltrrd 2866	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
14		df-br 5106	. . . . . 6 ⊢ (𝑀(𝐶 Func 𝐸)𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
15	13, 14	sylibr 237	. . . . 5 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐸)𝑁)
16	15	func1st 49706	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
17	11, 16	eqtrd 2800	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = 𝑀)
18	17	fveq1d 6873	. 2 ⊢ (𝜑 → ((1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))‘𝑋) = (𝑀‘𝑋))
19	5	func1st 49706	. . 3 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
20	2	func1st 49706	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
21	20	fveq1d 6873	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
22	19, 21	fveq12d 6878	. 2 ⊢ (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)‘((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)) = (𝐾‘(𝐹‘𝑋)))
23	9, 18, 22	3eqtr3rd 2809	1 ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = (𝑀‘𝑋))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ⟨cop 4591   class class class wbr 5105  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  Basecbs 17259   Func cfunc 17901   ∘_func ccofu 17903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  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 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-cat 17714  df-cid 17715  df-func 17905  df-cofu 17907

This theorem is referenced by:  uptrlem1  49839  uptrlem3  49841  uptr2  49850