Theorem homaval 17956

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 7356	. 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 17954	. . 3 ⊢ (𝜑 → 𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽‘𝑧))))
7		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
8	7	sneqd 4591	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → {𝑧} = {⟨𝑋, 𝑌⟩})
9	7	fveq2d 6830	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽‘𝑧) = (𝐽‘⟨𝑋, 𝑌⟩))
10		df-ov 7356	. . . . 5 ⊢ (𝑋𝐽𝑌) = (𝐽‘⟨𝑋, 𝑌⟩)
11	9, 10	eqtr4di 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽‘𝑧) = (𝑋𝐽𝑌))
12	8, 11	xpeq12d 5654	. . 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 5662	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
16		snex 5378	. . . . 5 ⊢ {⟨𝑋, 𝑌⟩} ∈ V
17		ovex 7386	. . . . 5 ⊢ (𝑋𝐽𝑌) ∈ V
18	16, 17	xpex 7693	. . . 4 ⊢ ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V
19	18	a1i 11	. . 3 ⊢ (𝜑 → ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V)
20	6, 12, 15, 19	fvmptd 6941	. 2 ⊢ (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
21	1, 20	eqtrid 2776	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  {csn 4579  ⟨cop 4585   × cxp 5621  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  Hom chom 17190  Catccat 17588  Hom_achoma 17948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-homa 17951

This theorem is referenced by:  elhoma  17957