Theorem termcid2 49980

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 49979	. 2 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))
9	1, 2, 3, 4	termcbasmo 49976	. . 3 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	fveq2d 6840	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘𝑌))
11	8, 10	eqtrd 2772	1 ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑌))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6494  (class class class)co 7362  Basecbs 17174  Hom chom 17226  Idccid 17626  TermCatctermc 49965

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 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5372

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-cat 17629  df-cid 17630  df-thinc 49911  df-termc 49966

This theorem is referenced by: (None)