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Theorem homafval 17160
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)
homafval.j 𝐽 = (Hom ‘𝐶)
Assertion
Ref Expression
homafval (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝐻(𝑥)   𝐽(𝑥)

Proof of Theorem homafval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 homarcl.h . 2 𝐻 = (Homa𝐶)
2 homafval.c . . 3 (𝜑𝐶 ∈ Cat)
3 fveq2 6497 . . . . . . 7 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 homafval.b . . . . . . 7 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2827 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
65sqxpeqd 5436 . . . . 5 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
7 fveq2 6497 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8 homafval.j . . . . . . . 8 𝐽 = (Hom ‘𝐶)
97, 8syl6eqr 2827 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽)
109fveq1d 6499 . . . . . 6 (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽𝑥))
1110xpeq2d 5434 . . . . 5 (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽𝑥)))
126, 11mpteq12dv 5009 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
13 df-homa 17157 . . . 4 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
144fvexi 6511 . . . . . 6 𝐵 ∈ V
1514, 14xpex 7292 . . . . 5 (𝐵 × 𝐵) ∈ V
1615mptex 6811 . . . 4 (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))) ∈ V
1712, 13, 16fvmpt 6594 . . 3 (𝐶 ∈ Cat → (Homa𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
182, 17syl 17 . 2 (𝜑 → (Homa𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
191, 18syl5eq 2821 1 (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  wcel 2051  {csn 4436  cmpt 5005   × cxp 5402  cfv 6186  Basecbs 16338  Hom chom 16431  Catccat 16806  Homachoma 17154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-homa 17157
This theorem is referenced by:  homaf  17161  homaval  17162
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