Theorem cofu1stfix 50417

Description: Post-composition by an object-fixing functor preserves the entire object function, including its exact domain. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)

Hypotheses

Ref	Expression
cofu1stfix.a	⊢ 𝐴 = (Base‘𝐸)
cofu1stfix.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
cofu1stfix.k	⊢ (𝜑 → 𝐾 ∈ (𝐸 Func 𝐸))
cofu1stfix.i	⊢ (𝜑 → (1st ‘𝐾) = ( I ↾ 𝐴))

Assertion

Ref	Expression
cofu1stfix	⊢ (𝜑 → (1st ‘(𝐾 ∘func 𝐹)) = (1st ‘𝐹))

Proof of Theorem cofu1stfix

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		cofu1stfix.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
3		cofu1stfix.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐸 Func 𝐸))
4	1, 2, 3	cofu1st 18087	. 2 ⊢ (𝜑 → (1st ‘(𝐾 ∘func 𝐹)) = ((1st ‘𝐾) ∘ (1st ‘𝐹)))
5		cofu1stfix.i	. . 3 ⊢ (𝜑 → (1st ‘𝐾) = ( I ↾ 𝐴))
6	5	coeq1d 5845	. 2 ⊢ (𝜑 → ((1st ‘𝐾) ∘ (1st ‘𝐹)) = (( I ↾ 𝐴) ∘ (1st ‘𝐹)))
7		cofu1stfix.a	. . . 4 ⊢ 𝐴 = (Base‘𝐸)
8	2	func1st2nd 50395	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
9	1, 7, 8	funcf1 18070	. . 3 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐷)⟶𝐴)
10		fcoi2 6779	. . 3 ⊢ ((1st ‘𝐹):(Base‘𝐷)⟶𝐴 → (( I ↾ 𝐴) ∘ (1st ‘𝐹)) = (1st ‘𝐹))
11	9, 10	syl 18	. 2 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ (1st ‘𝐹)) = (1st ‘𝐹))
12	4, 6, 11	3eqtrd 2802	1 ⊢ (𝜑 → (1st ‘(𝐾 ∘func 𝐹)) = (1st ‘𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   I cid 5549   ↾ cres 5657   ∘ ccom 5659  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024  Basecbs 17416   Func cfunc 18058   ∘_func ccofu 18060

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-map 8877  df-ixp 8954  df-func 18062  df-cofu 18064

This theorem is used by:  fucoaltfunc  50696