Theorem termcnatval 49998

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 17910	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩𝑁⟨(1st ‘𝐺), (2nd ‘𝐺)⟩))
4		termcnatval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5	1, 3, 4	natfn 17913	. . . 4 ⊢ (𝜑 → 𝐴 Fn 𝐵)
6		termcnatval.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ TermCat)
7		termcnatval.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	6, 4, 7	termcbas2 49945	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑋})
9	8	fneq2d 6581	. . . 4 ⊢ (𝜑 → (𝐴 Fn 𝐵 ↔ 𝐴 Fn {𝑋}))
10	5, 9	mpbid 232	. . 3 ⊢ (𝜑 → 𝐴 Fn {𝑋})
11		fnsnbg 7108	. . . 4 ⊢ (𝑋 ∈ 𝐵 → (𝐴 Fn {𝑋} ↔ 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩}))
12	7, 11	syl 17	. . 3 ⊢ (𝜑 → (𝐴 Fn {𝑋} ↔ 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩}))
13	10, 12	mpbid 232	. 2 ⊢ (𝜑 → 𝐴 = {⟨𝑋, (𝐴‘𝑋)⟩})
14		termcnatval.r	. . . 4 ⊢ 𝑅 = (𝐴‘𝑋)
15	14	opeq2i 4810	. . 3 ⊢ ⟨𝑋, 𝑅⟩ = ⟨𝑋, (𝐴‘𝑋)⟩
16	15	sneqi 4568	. 2 ⊢ {⟨𝑋, 𝑅⟩} = {⟨𝑋, (𝐴‘𝑋)⟩}
17	13, 16	eqtr4di 2788	1 ⊢ (𝜑 → 𝐴 = {⟨𝑋, 𝑅⟩})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {csn 4557  ⟨cop 4563   Fn wfn 6482  ‘cfv 6487  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17168   Nat cnat 17900  TermCatctermc 49935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-ixp 8835  df-func 17814  df-nat 17902  df-termc 49936

This theorem is referenced by:  diag2f1olem  49999