Theorem termcnatval 50643

Description: Value of natural transformations for a terminal category. (Contributed by Zhi Wang, 21-Oct-2025.)

Hypotheses

Ref	Expression
termcnatval.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcnatval.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
termcnatval.a	⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
termcnatval.b	⊢ 𝐵 = (Base‘𝐶)
termcnatval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
termcnatval.r	⊢ 𝑅 = (𝐴‘𝑋)

Assertion

Ref	Expression
termcnatval	⊢ (𝜑 → 𝐴 = {⟨𝑋, 𝑅⟩})

Proof of Theorem termcnatval

Step	Hyp	Ref	Expression
1		termcnatval.n	. . . . 5 ⊢ 𝑁 = (𝐶 Nat 𝐷)
2		termcnatval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
3	1, 2	nat1st2nd 18129	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
4		termcnatval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5	1, 3, 4	natfn 18132	. . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐵)
6		termcnatval.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ TermCat)
7		termcnatval.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	6, 4, 7	termcbas2 50590	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑋})
9	8	fneq2d 6633	. . . 4 ⊢ (𝜑 → (𝐴 Fn 𝐵 ↔ 𝐴 Fn {𝑋}))
10	5, 9	mpbid 235	. . 3 ⊢ (𝜑 → 𝐴 Fn {𝑋})
11		fnsnbg 7169	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (𝐴 Fn {𝑋} ↔ 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩}))
12	7, 11	syl 18	. . 3 ⊢ (𝜑 → (𝐴 Fn {𝑋} ↔ 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩}))
13	10, 12	mpbid 235	. 2 ⊢ (𝜑 → 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩})
14		termcnatval.r	. . . 4 ⊢ 𝑅 = (𝐴‘𝑋)
15	14	opeq2i 4836	. . 3 ⊢ ⟨𝑋, 𝑅⟩ = ⟨𝑋, (𝐴‘𝑋)⟩
16	15	sneqi 4594	. 2 ⊢ {⟨𝑋, 𝑅⟩} = {⟨𝑋, (𝐴‘𝑋)⟩}
17	13, 16	eqtr4di 2813	1 ⊢ (𝜑 → 𝐴 = {⟨𝑋, 𝑅⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144  {csn 4583  ⟨cop 4589   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17387   Nat cnat 18119  TermCatctermc 50580

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-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-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-ixp 8926  df-func 18033  df-nat 18121  df-termc 50581

This theorem is used by:  diag2f1olem  50644