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Theorem homffval 17643
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
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 6885 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3 homffval.b . . . . . 6 𝐵 = (Base‘𝐶)
42, 3eqtr4di 2784 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5 fveq2 6885 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6 homffval.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
75, 6eqtr4di 2784 . . . . . 6 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
87oveqd 7422 . . . . 5 (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
94, 4, 8mpoeq123dv 7480 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
10 df-homf 17623 . . . 4 Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
113fvexi 6899 . . . . 5 𝐵 ∈ V
1211, 11mpoex 8065 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) ∈ V
139, 10, 12fvmpt 6992 . . 3 (𝐶 ∈ V → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
14 fvprc 6877 . . . 4 𝐶 ∈ V → (Homf𝐶) = ∅)
15 fvprc 6877 . . . . . . 7 𝐶 ∈ V → (Base‘𝐶) = ∅)
163, 15eqtrid 2778 . . . . . 6 𝐶 ∈ V → 𝐵 = ∅)
1716olcd 871 . . . . 5 𝐶 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
18 0mpo0 7488 . . . . 5 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
1917, 18syl 17 . . . 4 𝐶 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
2014, 19eqtr4d 2769 . . 3 𝐶 ∈ V → (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
2113, 20pm2.61i 182 . 2 (Homf𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
221, 21eqtri 2754 1 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 844   = wceq 1533  wcel 2098  Vcvv 3468  c0 4317  cfv 6537  (class class class)co 7405  cmpo 7407  Basecbs 17153  Hom chom 17217  Homf chomf 17619
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-homf 17623
This theorem is referenced by:  fnhomeqhomf  17644  homfval  17645  homffn  17646  homfeq  17647  oppchomf  17675  reschomf  17788
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