Theorem termcfunc1eq 50875

Description: Functors with terminal source are equal when they agree on its object. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
termcfunc1eq.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcfunc1eq.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
termcfunc1eq.b	⊢ 𝐵 = (Base‘𝐶)
termcfunc1eq.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
termcfunc1eq.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
termcfunc1eq.e	⊢ (𝜑 → ((1st ‘𝐹)‘𝑌) = ((1st ‘𝐺)‘𝑌))

Assertion

Ref	Expression
termcfunc1eq	⊢ (𝜑 → 𝐹 = 𝐺)

Proof of Theorem termcfunc1eq

Step	Hyp	Ref	Expression
1		termcfunc1eq.e	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑌) = ((1st ‘𝐺)‘𝑌))
2	1	opeq2d 4850	. . . 4 ⊢ (𝜑 → ⟨𝑌, ((1st ‘𝐹)‘𝑌)⟩ = ⟨𝑌, ((1st ‘𝐺)‘𝑌)⟩)
3	2	sneqd 4596	. . 3 ⊢ (𝜑 → {⟨𝑌, ((1st ‘𝐹)‘𝑌)⟩} = {⟨𝑌, ((1st ‘𝐺)‘𝑌)⟩})
4	1	fveq2d 6912	. . . . . . 7 ⊢ (𝜑 → ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌)) = ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌)))
5	4	opeq2d 4850	. . . . . 6 ⊢ (𝜑 → ⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌))⟩ = ⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌))⟩)
6	5	sneqd 4596	. . . . 5 ⊢ (𝜑 → {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌))⟩} = {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌))⟩})
7	6	opeq2d 4850	. . . 4 ⊢ (𝜑 → ⟨⟨𝑌, 𝑌⟩, {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌))⟩}⟩ = ⟨⟨𝑌, 𝑌⟩, {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌))⟩}⟩)
8	7	sneqd 4596	. . 3 ⊢ (𝜑 → {⟨⟨𝑌, 𝑌⟩, {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌))⟩}⟩} = {⟨⟨𝑌, 𝑌⟩, {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌))⟩}⟩})
9	3, 8	opeq12d 4851	. 2 ⊢ (𝜑 → ⟨{⟨𝑌, ((1st ‘𝐹)‘𝑌)⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌))⟩}⟩}⟩ = ⟨{⟨𝑌, ((1st ‘𝐺)‘𝑌)⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌))⟩}⟩}⟩)
10		eqid 2763	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
11		termcfunc1eq.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ TermCat)
12		termcfunc1eq.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
13		termcfunc1eq.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
14		termcfunc1eq.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
15		eqid 2763	. . . 4 ⊢ ((1st ‘𝐹)‘𝑌) = ((1st ‘𝐹)‘𝑌)
16		eqid 2763	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
17		eqid 2763	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
18	10, 11, 12, 13, 14, 15, 16, 17	termcfuncval 50872	. . 3 ⊢ (𝜑 → (((1st ‘𝐹)‘𝑌) ∈ (Base‘𝐷) ∧ 𝐹 = ⟨{⟨𝑌, ((1st ‘𝐹)‘𝑌)⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌))⟩}⟩}⟩))
19	18	simprd 501	. 2 ⊢ (𝜑 → 𝐹 = ⟨{⟨𝑌, ((1st ‘𝐹)‘𝑌)⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌))⟩}⟩}⟩)
20		termcfunc1eq.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
21		eqid 2763	. . . 4 ⊢ ((1st ‘𝐺)‘𝑌) = ((1st ‘𝐺)‘𝑌)
22	10, 11, 20, 13, 14, 21, 16, 17	termcfuncval 50872	. . 3 ⊢ (𝜑 → (((1st ‘𝐺)‘𝑌) ∈ (Base‘𝐷) ∧ 𝐺 = ⟨{⟨𝑌, ((1st ‘𝐺)‘𝑌)⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌))⟩}⟩}⟩))
23	22	simprd 501	. 2 ⊢ (𝜑 → 𝐺 = ⟨{⟨𝑌, ((1st ‘𝐺)‘𝑌)⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌))⟩}⟩}⟩)
24	9, 19, 23	3eqtr4d 2808	1 ⊢ (𝜑 → 𝐹 = 𝐺)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {csn 4588  ⟨cop 4838  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  Basecbs 17416  Idccid 17868   Func cfunc 18058  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-thinc 50758  df-termc 50813

This theorem is used by:  termccofix  50876