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Theorem homffval 16955
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homffval.f 𝐹 = (Homf𝐶)
homffval.b 𝐵 = (Base‘𝐶)
homffval.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
homffval 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐻,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem homffval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 homffval.f . 2 𝐹 = (Homf𝐶)
2 fveq2 6669 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 homffval.b . . . . . 6 𝐵 = (Base‘𝐶)
42, 3syl6eqr 2879 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fveq2 6669 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6 homffval.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
75, 6syl6eqr 2879 . . . . . 6 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
87oveqd 7167 . . . . 5 (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
94, 4, 8mpoeq123dv 7223 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
10 df-homf 16936 . . . 4 Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
113fvexi 6683 . . . . 5 𝐵 ∈ V
1211, 11mpoex 7773 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) ∈ V
139, 10, 12fvmpt 6767 . . 3 (𝐶 ∈ V → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
14 mpo0 7233 . . . . 5 (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦)) = ∅
1514eqcomi 2835 . . . 4 ∅ = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦))
16 fvprc 6662 . . . 4 𝐶 ∈ V → (Homf𝐶) = ∅)
17 fvprc 6662 . . . . . 6 𝐶 ∈ V → (Base‘𝐶) = ∅)
183, 17syl5eq 2873 . . . . 5 𝐶 ∈ V → 𝐵 = ∅)
19 mpoeq12 7221 . . . . 5 ((𝐵 = ∅ ∧ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦)))
2018, 18, 19syl2anc 584 . . . 4 𝐶 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦)))
2115, 16, 203eqtr4a 2887 . . 3 𝐶 ∈ V → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
2213, 21pm2.61i 183 . 2 (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
231, 22eqtri 2849 1 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  Vcvv 3500  c0 4295  cfv 6354  (class class class)co 7150  cmpo 7152  Basecbs 16478  Hom chom 16571  Homf chomf 16932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-homf 16936
This theorem is referenced by:  fnhomeqhomf  16956  homfval  16957  homffn  16958  homfeq  16959  oppchomf  16985  reschomf  17096
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