Theorem termccofix 50876

Description: Post-composition by an object-fixing endofunctor leaves a functor with terminal source literally unchanged. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
termcfunc1eq.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcfunc1eq.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
termccofix.s	⊢ (𝜑 → 𝑆 ∈ (𝐷 Func 𝐷))
termccofix.i	⊢ (𝜑 → (1st ‘𝑆) = ( I ↾ 𝐴))

Assertion

Ref	Expression
termccofix	⊢ (𝜑 → (𝑆 ∘func 𝐹) = 𝐹)

Proof of Theorem termccofix

Step	Hyp	Ref	Expression
1		termcfunc1eq.c	. 2 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcfunc1eq.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		termccofix.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ (𝐷 Func 𝐷))
4	2, 3	cofucl 18092	. 2 ⊢ (𝜑 → (𝑆 ∘func 𝐹) ∈ (𝐶 Func 𝐷))
5		eqid 2763	. 2 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	1, 5	termco 50821	. 2 ⊢ (𝜑 → ∪ (Base‘𝐶) ∈ (Base‘𝐶))
7	5, 2, 3, 6	cofu1 18088	. . 3 ⊢ (𝜑 → ((1st ‘(𝑆 ∘func 𝐹))‘∪ (Base‘𝐶)) = ((1st ‘𝑆)‘((1st ‘𝐹)‘∪ (Base‘𝐶))))
8		termccofix.i	. . . 4 ⊢ (𝜑 → (1st ‘𝑆) = ( I ↾ 𝐴))
9	8	fveq1d 6910	. . 3 ⊢ (𝜑 → ((1st ‘𝑆)‘((1st ‘𝐹)‘∪ (Base‘𝐶))) = (( I ↾ 𝐴)‘((1st ‘𝐹)‘∪ (Base‘𝐶))))
10		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
11	2	func1st2nd 50395	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
12	5, 10, 11	funcf1 18070	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
13	12, 6	ffvelcdmd 7109	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐹)‘∪ (Base‘𝐶)) ∈ (Base‘𝐷))
14	8	dmeqd 5894	. . . . . . 7 ⊢ (𝜑 → dom (1st ‘𝑆) = dom ( I ↾ 𝐴))
15		dmresi 6056	. . . . . . 7 ⊢ dom ( I ↾ 𝐴) = 𝐴
16	14, 15	eqtrdi 2814	. . . . . 6 ⊢ (𝜑 → dom (1st ‘𝑆) = 𝐴)
17	3	func1st2nd 50395	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑆)(𝐷 Func 𝐷)(2nd ‘𝑆))
18	10, 10, 17	funcf1 18070	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑆):(Base‘𝐷)⟶(Base‘𝐷))
19	18	fdmd 6742	. . . . . 6 ⊢ (𝜑 → dom (1st ‘𝑆) = (Base‘𝐷))
20	16, 19	eqtr3d 2800	. . . . 5 ⊢ (𝜑 → 𝐴 = (Base‘𝐷))
21	13, 20	eleqtrrd 2867	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹)‘∪ (Base‘𝐶)) ∈ 𝐴)
22		fvresi 7201	. . . 4 ⊢ (((1st ‘𝐹)‘∪ (Base‘𝐶)) ∈ 𝐴 → (( I ↾ 𝐴)‘((1st ‘𝐹)‘∪ (Base‘𝐶))) = ((1st ‘𝐹)‘∪ (Base‘𝐶)))
23	21, 22	syl 18	. . 3 ⊢ (𝜑 → (( I ↾ 𝐴)‘((1st ‘𝐹)‘∪ (Base‘𝐶))) = ((1st ‘𝐹)‘∪ (Base‘𝐶)))
24	7, 9, 23	3eqtrd 2802	. 2 ⊢ (𝜑 → ((1st ‘(𝑆 ∘func 𝐹))‘∪ (Base‘𝐶)) = ((1st ‘𝐹)‘∪ (Base‘𝐶)))
25	1, 4, 5, 2, 6, 24	termcfunc1eq 50875	1 ⊢ (𝜑 → (𝑆 ∘func 𝐹) = 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∪ cuni 4870   I cid 5549  dom cdm 5655   ↾ cres 5657  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024  Basecbs 17416   Func cfunc 18058   ∘_func ccofu 18060  TermCatctermc 50812

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-rmo 3369  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-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-map 8877  df-ixp 8954  df-cat 17871  df-cid 17872  df-func 18062  df-cofu 18064  df-thinc 50758  df-termc 50813

This theorem is used by:  termcpostfix  50877