Theorem cofu1a 49581

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 5087	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4	2, 3	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
5		cofu1a.k	. . . 4 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
6		df-br 5087	. . . 4 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
7	5, 6	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
8		cofu1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9	1, 4, 7, 8	cofu1 17842	. 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 6838	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = (1st ‘⟨𝑀, 𝑁⟩))
12	4, 7	cofucl 17846	. . . . . . 7 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
13	10, 12	eqeltrrd 2838	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
14		df-br 5087	. . . . . 6 ⊢ (𝑀(𝐶 Func 𝐸)𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ (𝐶 Func 𝐸))
15	13, 14	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐸)𝑁)
16	15	func1st 49564	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
17	11, 16	eqtrd 2772	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩)) = 𝑀)
18	17	fveq1d 6836	. 2 ⊢ (𝜑 → ((1st ‘(⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))‘𝑋) = (𝑀‘𝑋))
19	5	func1st 49564	. . 3 ⊢ (𝜑 → (1st ‘⟨𝐾, 𝐿⟩) = 𝐾)
20	2	func1st 49564	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
21	20	fveq1d 6836	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹‘𝑋))
22	19, 21	fveq12d 6841	. 2 ⊢ (𝜑 → ((1st ‘⟨𝐾, 𝐿⟩)‘((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)) = (𝐾‘(𝐹‘𝑋)))
23	9, 18, 22	3eqtr3rd 2781	1 ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = (𝑀‘𝑋))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4574   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  Basecbs 17170   Func cfunc 17812   ∘_func ccofu 17814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8768  df-ixp 8839  df-cat 17625  df-cid 17626  df-func 17816  df-cofu 17818

This theorem is referenced by:  uptrlem1  49697  uptrlem3  49699  uptr2  49708