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Theorem homafval 16880
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 6352 . . . . . . 7 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 homafval.b . . . . . . 7 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2812 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
65sqxpeqd 5298 . . . . 5 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
7 fveq2 6352 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8 homafval.j . . . . . . . 8 𝐽 = (Hom ‘𝐶)
97, 8syl6eqr 2812 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽)
109fveq1d 6354 . . . . . 6 (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽𝑥))
1110xpeq2d 5296 . . . . 5 (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽𝑥)))
126, 11mpteq12dv 4885 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
13 df-homa 16877 . . . 4 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
14 fvex 6362 . . . . . . 7 (Base‘𝐶) ∈ V
154, 14eqeltri 2835 . . . . . 6 𝐵 ∈ V
1615, 15xpex 7127 . . . . 5 (𝐵 × 𝐵) ∈ V
1716mptex 6650 . . . 4 (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))) ∈ V
1812, 13, 17fvmpt 6444 . . 3 (𝐶 ∈ Cat → (Homa𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
192, 18syl 17 . 2 (𝜑 → (Homa𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
201, 19syl5eq 2806 1 (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321  cmpt 4881   × cxp 5264  cfv 6049  Basecbs 16059  Hom chom 16154  Catccat 16526  Homachoma 16874
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 7114
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-homa 16877
This theorem is referenced by:  homaf  16881  homaval  16882
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