Theorem homaval 18199

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 7421	. 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 18197	. . 3 ⊢ (𝜑 → 𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽‘𝑧))))
7		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
8	7	sneqd 4596	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → {𝑧} = {⟨𝑋, 𝑌⟩})
9	7	fveq2d 6887	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽‘𝑧) = (𝐽‘⟨𝑋, 𝑌⟩))
10		df-ov 7421	. . . . 5 ⊢ (𝑋𝐽𝑌) = (𝐽‘⟨𝑋, 𝑌⟩)
11	9, 10	eqtr4di 2814	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽‘𝑧) = (𝑋𝐽𝑌))
12	8, 11	xpeq12d 5682	. . 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 5690	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
16		snex 5397	. . . . 5 ⊢ {⟨𝑋, 𝑌⟩} ∈ V
17		ovex 7451	. . . . 5 ⊢ (𝑋𝐽𝑌) ∈ V
18	16, 17	xpex 7765	. . . 4 ⊢ ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V
19	18	a1i 11	. . 3 ⊢ (𝜑 → ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V)
20	6, 12, 15, 19	fvmptd 6999	. 2 ⊢ (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
21	1, 20	eqtrid 2808	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  {csn 4584  ⟨cop 4590   × cxp 5649  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  Hom chom 17432  Catccat 17831  Hom_achoma 18191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-homa 18194

This theorem is used by:  elhoma  18200