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Theorem homaval 16452
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 6529 . 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 16450 . . 3 (𝜑𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽𝑧))))
7 simpr 475 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
87sneqd 4136 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → {𝑧} = {⟨𝑋, 𝑌⟩})
97fveq2d 6091 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽𝑧) = (𝐽‘⟨𝑋, 𝑌⟩))
10 df-ov 6529 . . . . 5 (𝑋𝐽𝑌) = (𝐽‘⟨𝑋, 𝑌⟩)
119, 10syl6eqr 2661 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽𝑧) = (𝑋𝐽𝑌))
128, 11xpeq12d 5053 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ({𝑧} × (𝐽𝑧)) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
13 homaval.x . . . 4 (𝜑𝑋𝐵)
14 homaval.y . . . 4 (𝜑𝑌𝐵)
15 opelxpi 5061 . . . 4 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
1613, 14, 15syl2anc 690 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
17 snex 4829 . . . . 5 {⟨𝑋, 𝑌⟩} ∈ V
18 ovex 6554 . . . . 5 (𝑋𝐽𝑌) ∈ V
1917, 18xpex 6837 . . . 4 ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V
2019a1i 11 . . 3 (𝜑 → ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V)
216, 12, 16, 20fvmptd 6181 . 2 (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
221, 21syl5eq 2655 1 (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  {csn 4124  cop 4130   × cxp 5025  cfv 5789  (class class class)co 6526  Basecbs 15643  Hom chom 15727  Catccat 16096  Homachoma 16444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-homa 16447
This theorem is referenced by:  elhoma  16453
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