Theorem termcid2 49989

Description: The morphism of a terminal category is an identity morphism. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)
termcbasmo.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
termcbasmo.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
termcid.h	⊢ 𝐻 = (Hom ‘𝐶)
termcid.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
termcid.i	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
termcid2	⊢ (𝜑 → 𝐹 = ( 1 ‘𝑌))

Proof of Theorem termcid2

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		termcbasmo.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4		termcbasmo.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
5		termcid.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
6		termcid.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
7		termcid.i	. . 3 ⊢ 1 = (Id‘𝐶)
8	1, 2, 3, 4, 5, 6, 7	termcid 49988	. 2 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))
9	1, 2, 3, 4	termcbasmo 49985	. . 3 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	fveq2d 6834	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘𝑌))
11	8, 10	eqtrd 2776	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑌))

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  Hom chom 17226  Idccid 17626  TermCatctermc 49974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  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 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-cat 17629  df-cid 17630  df-thinc 49920  df-termc 49975

This theorem is referenced by: (None)