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Theorem homaval 16902
 Description: Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homarcl.h 𝐻 = (Homa𝐶)
homafval.b 𝐵 = (Base‘𝐶)
homafval.c (𝜑𝐶 ∈ Cat)
homaval.j 𝐽 = (Hom ‘𝐶)
homaval.x (𝜑𝑋𝐵)
homaval.y (𝜑𝑌𝐵)
Assertion
Ref Expression
homaval (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))

Proof of Theorem homaval
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ov 6817 . 2 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2 homarcl.h . . . 4 𝐻 = (Homa𝐶)
3 homafval.b . . . 4 𝐵 = (Base‘𝐶)
4 homafval.c . . . 4 (𝜑𝐶 ∈ Cat)
5 homaval.j . . . 4 𝐽 = (Hom ‘𝐶)
62, 3, 4, 5homafval 16900 . . 3 (𝜑𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽𝑧))))
7 simpr 479 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
87sneqd 4333 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → {𝑧} = {⟨𝑋, 𝑌⟩})
97fveq2d 6357 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽𝑧) = (𝐽‘⟨𝑋, 𝑌⟩))
10 df-ov 6817 . . . . 5 (𝑋𝐽𝑌) = (𝐽‘⟨𝑋, 𝑌⟩)
119, 10syl6eqr 2812 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽𝑧) = (𝑋𝐽𝑌))
128, 11xpeq12d 5297 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ({𝑧} × (𝐽𝑧)) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
13 homaval.x . . . 4 (𝜑𝑋𝐵)
14 homaval.y . . . 4 (𝜑𝑌𝐵)
15 opelxpi 5305 . . . 4 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
1613, 14, 15syl2anc 696 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
17 snex 5057 . . . . 5 {⟨𝑋, 𝑌⟩} ∈ V
18 ovex 6842 . . . . 5 (𝑋𝐽𝑌) ∈ V
1917, 18xpex 7128 . . . 4 ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V
2019a1i 11 . . 3 (𝜑 → ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V)
216, 12, 16, 20fvmptd 6451 . 2 (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
221, 21syl5eq 2806 1 (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340  {csn 4321  ⟨cop 4327   × cxp 5264  ‘cfv 6049  (class class class)co 6814  Basecbs 16079  Hom chom 16174  Catccat 16546  Homachoma 16894 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-homa 16897 This theorem is referenced by:  elhoma  16903
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