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Theorem homafval 17991
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 6885 . . . . . . 7 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 homafval.b . . . . . . 7 𝐵 = (Base‘𝐶)
53, 4eqtr4di 2784 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
65sqxpeqd 5701 . . . . 5 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
7 fveq2 6885 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8 homafval.j . . . . . . . 8 𝐽 = (Hom ‘𝐶)
97, 8eqtr4di 2784 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐽)
109fveq1d 6887 . . . . . 6 (𝑐 = 𝐶 → ((Hom ‘𝑐)‘𝑥) = (𝐽𝑥))
1110xpeq2d 5699 . . . . 5 (𝑐 = 𝐶 → ({𝑥} × ((Hom ‘𝑐)‘𝑥)) = ({𝑥} × (𝐽𝑥)))
126, 11mpteq12dv 5232 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
13 df-homa 17988 . . . 4 Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
144fvexi 6899 . . . . . 6 𝐵 ∈ V
1514, 14xpex 7737 . . . . 5 (𝐵 × 𝐵) ∈ V
1615mptex 7220 . . . 4 (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))) ∈ V
1712, 13, 16fvmpt 6992 . . 3 (𝐶 ∈ Cat → (Homa𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
182, 17syl 17 . 2 (𝜑 → (Homa𝐶) = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
191, 18eqtrid 2778 1 (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {csn 4623  cmpt 5224   × cxp 5667  cfv 6537  Basecbs 17153  Hom chom 17217  Catccat 17617  Homachoma 17985
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-homa 17988
This theorem is referenced by:  homaf  17992  homaval  17993
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