Theorem termcfunc1eq 50761

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 4839	. . . 4 ⊢ (𝜑 → ⟨𝑌, ((1st ‘𝐹)‘𝑌)⟩ = ⟨𝑌, ((1st ‘𝐺)‘𝑌)⟩)
3	2	sneqd 4595	. . 3 ⊢ (𝜑 → {⟨𝑌, ((1st ‘𝐹)‘𝑌)⟩} = {⟨𝑌, ((1st ‘𝐺)‘𝑌)⟩})
4	1	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌)) = ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌)))
5	4	opeq2d 4839	. . . . . 6 ⊢ (𝜑 → ⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌))⟩ = ⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌))⟩)
6	5	sneqd 4595	. . . . 5 ⊢ (𝜑 → {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐹)‘𝑌))⟩} = {⟨((Id‘𝐶)‘𝑌), ((Id‘𝐷)‘((1st ‘𝐺)‘𝑌))⟩})
7	6	opeq2d 4839	. . . 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 4595	. . 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 4840	. 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 2760	. . . 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 2760	. . . 4 ⊢ ((1st ‘𝐹)‘𝑌) = ((1st ‘𝐹)‘𝑌)
16		eqid 2760	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
17		eqid 2760	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
18	10, 11, 12, 13, 14, 15, 16, 17	termcfuncval 50758	. . 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 2760	. . . 4 ⊢ ((1st ‘𝐺)‘𝑌) = ((1st ‘𝐺)‘𝑌)
22	10, 11, 20, 13, 14, 21, 16, 17	termcfuncval 50758	. . 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 2805	1 ⊢ (𝜑 → 𝐹 = 𝐺)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {csn 4583  ⟨cop 4589  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  Basecbs 17389  Idccid 17841   Func cfunc 18031  TermCatctermc 50698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8851  df-ixp 8928  df-cat 17844  df-cid 17845  df-func 18035  df-thinc 50644  df-termc 50699

This theorem is used by:  termccofix  50762