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Theorem homffval 16571
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 6353 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 homffval.b . . . . . 6 𝐵 = (Base‘𝐶)
42, 3syl6eqr 2812 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fveq2 6353 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6 homffval.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
75, 6syl6eqr 2812 . . . . . 6 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
87oveqd 6831 . . . . 5 (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
94, 4, 8mpt2eq123dv 6883 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
10 df-homf 16552 . . . 4 Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
11 fvex 6363 . . . . . 6 (Base‘𝐶) ∈ V
123, 11eqeltri 2835 . . . . 5 𝐵 ∈ V
1312, 12mpt2ex 7416 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) ∈ V
149, 10, 13fvmpt 6445 . . 3 (𝐶 ∈ V → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
15 mpt20 6891 . . . . 5 (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦)) = ∅
1615eqcomi 2769 . . . 4 ∅ = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦))
17 fvprc 6347 . . . 4 𝐶 ∈ V → (Homf𝐶) = ∅)
18 fvprc 6347 . . . . . 6 𝐶 ∈ V → (Base‘𝐶) = ∅)
193, 18syl5eq 2806 . . . . 5 𝐶 ∈ V → 𝐵 = ∅)
20 mpt2eq12 6881 . . . . 5 ((𝐵 = ∅ ∧ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦)))
2119, 19, 20syl2anc 696 . . . 4 𝐶 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥𝐻𝑦)))
2216, 17, 213eqtr4a 2820 . . 3 𝐶 ∈ V → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
2314, 22pm2.61i 176 . 2 (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
241, 23eqtri 2782 1 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  cfv 6049  (class class class)co 6814  cmpt2 6816  Basecbs 16079  Hom chom 16174  Homf chomf 16548
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 7115
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-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-homf 16552
This theorem is referenced by:  fnhomeqhomf  16572  homfval  16573  homffn  16574  homfeq  16575  oppchomf  16601  reschomf  16712
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