Theorem homaval 17662

Description: Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
homarcl.h	⊢ 𝐻 = (Homa‘𝐶)
homafval.b	⊢ 𝐵 = (Base‘𝐶)
homafval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
homaval.j	⊢ 𝐽 = (Hom ‘𝐶)
homaval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
homaval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
homaval	⊢ (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))

Proof of Theorem homaval

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7258	. 2 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2		homarcl.h	. . . 4 ⊢ 𝐻 = (Homa‘𝐶)
3		homafval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4		homafval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		homaval.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐶)
6	2, 3, 4, 5	homafval 17660	. . 3 ⊢ (𝜑 → 𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽‘𝑧))))
7		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
8	7	sneqd 4570	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → {𝑧} = {⟨𝑋, 𝑌⟩})
9	7	fveq2d 6760	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽‘𝑧) = (𝐽‘⟨𝑋, 𝑌⟩))
10		df-ov 7258	. . . . 5 ⊢ (𝑋𝐽𝑌) = (𝐽‘⟨𝑋, 𝑌⟩)
11	9, 10	eqtr4di 2797	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽‘𝑧) = (𝑋𝐽𝑌))
12	8, 11	xpeq12d 5611	. . 3 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ({𝑧} × (𝐽‘𝑧)) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
13		homaval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14		homaval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
15	13, 14	opelxpd 5618	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
16		snex 5349	. . . . 5 ⊢ {⟨𝑋, 𝑌⟩} ∈ V
17		ovex 7288	. . . . 5 ⊢ (𝑋𝐽𝑌) ∈ V
18	16, 17	xpex 7581	. . . 4 ⊢ ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V
19	18	a1i 11	. . 3 ⊢ (𝜑 → ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V)
20	6, 12, 15, 19	fvmptd 6864	. 2 ⊢ (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
21	1, 20	eqtrid 2790	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558  ⟨cop 4564   × cxp 5578  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  Catccat 17290  Hom_achoma 17654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-homa 17657

This theorem is referenced by:  elhoma  17663