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Theorem homafval 16946
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 6375 . . . . . . 7 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 homafval.b . . . . . . 7 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2817 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
65sqxpeqd 5309 . . . . 5 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
7 fveq2 6375 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8 homafval.j . . . . . . . 8 𝐽 = (Hom ‘𝐶)
97, 8syl6eqr 2817 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽)
109fveq1d 6377 . . . . . 6 (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽𝑥))
1110xpeq2d 5307 . . . . 5 (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽𝑥)))
126, 11mpteq12dv 4892 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
13 df-homa 16943 . . . 4 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
144fvexi 6389 . . . . . 6 𝐵 ∈ V
1514, 14xpex 7160 . . . . 5 (𝐵 × 𝐵) ∈ V
1615mptex 6679 . . . 4 (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))) ∈ V
1712, 13, 16fvmpt 6471 . . 3 (𝐶 ∈ Cat → (Homa𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
182, 17syl 17 . 2 (𝜑 → (Homa𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
191, 18syl5eq 2811 1 (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  {csn 4334  cmpt 4888   × cxp 5275  cfv 6068  Basecbs 16132  Hom chom 16227  Catccat 16592  Homachoma 16940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-homa 16943
This theorem is referenced by:  homaf  16947  homaval  16948
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