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Theorem homafval 17301
 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 6655 . . . . . . 7 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 homafval.b . . . . . . 7 𝐵 = (Base‘𝐶)
53, 4eqtr4di 2851 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
65sqxpeqd 5555 . . . . 5 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
7 fveq2 6655 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8 homafval.j . . . . . . . 8 𝐽 = (Hom ‘𝐶)
97, 8eqtr4di 2851 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽)
109fveq1d 6657 . . . . . 6 (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽𝑥))
1110xpeq2d 5553 . . . . 5 (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽𝑥)))
126, 11mpteq12dv 5119 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
13 df-homa 17298 . . . 4 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
144fvexi 6669 . . . . . 6 𝐵 ∈ V
1514, 14xpex 7469 . . . . 5 (𝐵 × 𝐵) ∈ V
1615mptex 6973 . . . 4 (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))) ∈ V
1712, 13, 16fvmpt 6755 . . 3 (𝐶 ∈ Cat → (Homa𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
182, 17syl 17 . 2 (𝜑 → (Homa𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
191, 18syl5eq 2845 1 (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {csn 4528   ↦ cmpt 5114   × cxp 5521  ‘cfv 6332  Basecbs 16495  Hom chom 16588  Catccat 16947  Homachoma 17295 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-homa 17298 This theorem is referenced by:  homaf  17302  homaval  17303
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